The binomial expansion of (y - 3)⁴ is B) y⁴ - 12y³ + 54y² - 108y + 81.
What is the binomial theorem?The Binomial Theorem is the method of expanding an expression that has been raised to any finite power.
An expression of two terms having a degree n can be represented as,
(a - b)ⁿ = [tex]^nC_na^n - ^nC_{n-1}a^{n-1}b+^{n}C_{n-2}a^{n-2}b^2+...(-1)^n\times^nC_0a^0b^n[/tex].
Given, (y - 3)⁴ = [tex]^4C_4y^4.3^0 - ^4C_3y^3.3 + ^4C_2y^23^2-^4C_1y.3^3+^4C_0y^03^4[/tex].
(y - 3)⁴ = y⁴ - 12y³ + 54y² - 108y + 81.
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If 10400 dollars is invested at an interest rate of 8 percent per year, find the value of the investment at the
end of 5 years for the following compounding methods, to the nearest cent.
(a) Annual: $
(b) Semiannual: $
(c) Monthly: $
(d) Daily: $
The interest estimated by compounding methods are-
(a) Annual: n = 1; CI = $860
(b) Semiannual: n = 2; CI = $862.
(c) Monthly: n = 12; CI = $865
(d) Daily: n = 365; CI = $855
Explain the term compound interest?When you add the interest you have already earned back into the principal balance, you are earning compound interest, which increases your profits.The formula for computing the compound interest is -
CI = A - P
A = P(1 + r/100nt)∧rt
In which,
CI = compound interestP = principal ($10,400)n = number of times compoundedr = rate of interest (8%)t = time (5 year)(a) Annual: n = 1
A = 10,400(1 + 8/500)∧5
A = 10,400 x 1.08
A = 11,260
CI = 11,260 - 10,400
CI = $860
(b) Semiannual: n = 2
A = 10,400(1 + 8/1000)∧10
A = 10,400 x 1.089
A = 11,262
CI = $862.
(c) Monthly: n = 12
A = 10,400(1 + 8/6000)∧60
A = 10,400 x 1.083
A = 11,265
CI = $865
(d) Daily: n = 365
A = 10,400(1 + 8/182500)∧1825
A = 10,400 x 1.0832
A = 11,255
CI = $855
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ben wants to triple his spaghetti sauce recipe, which calls for 2 pound 8 ounces of ground turkey. how much ground turkey will he need?
He needs 2.5 pounds of ground turkey.
Given,
In the question:
Ben wants to triple his spaghetti sauce recipe, which calls for 2 pound 8 ounces of ground turkey.
To find the how much pounds ground turkey will he need?
Now, According to the question:
Let's know:
How do you convert ounces to pounds formula?
In other words, dividing a number of ounces by 16 will get you an answer in pounds. In our example, dividing the baby's weight in ounces by 16 gets us 160/16 = 10 lbs.
We know that there are 16 ounces in 1 pound,
It is meaning that 8 ounces is half a pound.
Adding this to the 2 pound gives 2 + 1/2 = 2.5 pounds of ground turkey.
Hence, 2.5 pounds of ground turkey.
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Determine whether the pairs of functions in Problems 20 through 26 are linearly independent or linearly dependent on the real line. 20. f(x) = g(x) = cos? x + sinx 21. f(x)=x,8() = x+|x| 23. f(x) = xe", g(x) = |x|e" 22. f(x) = 1 + x, 8(x) = 1 + x 24. f(x) = sinx, g(x) = 1 - cos 2x 25. f(x) = e sinx, g(x) = e* cos x 26. f(x) = 2 cos x + 3 sinx, g(x) = 3 cos x - 2 sin x
The pairs of function which are linearly independent are f(x) = 2cos x + 3 sin x and g(x) = 3cox x - 2 sin x.
We need to determine the pairs of functions are linearly independent or linearly dependent on the real line
f(x) = 2cos x + 3 sin x
g(x) = 3cox x - 2 sin x
Let c₁f(x) and c₂g(x) = 0
using wornkskin method
w(f, y) = [tex]\left[\begin{array}{cc}f(x)&g(x)\\f'(x)&g'(x)\\\end{array}\right] \\\left[\begin{array}{cc}2cosx + 3sin x&3cos x - 2 sin x\\-2sin x + 3 cos x&-3sin x - 2 cos x\\\end{array}\right][/tex]
= - (2cos x + 3sin x) (3 sinx + 2 cos x) - (3cos x - 2sin x) (3cos x - 2sin x)
= - (9sin²x + 4cos² x + 12cos x sin x) + 9cos² x + 4 sin² x - 12 sin x cos x
= - (13 sin² x + 13 cos² x )
= - 13
since,
w(f , g) ≠ 0
f, g are linearly independent
Therefore, the function f(x) = 2cos x + 3 sin x and g(x) = 3cox x - 2 sin x are linearly independent.
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The sum of the digits of a three-digit number is 13. The tens digit, t, is 1 more than the hundreds digit, h. The units digit, u, is 3 more than the sum of the tens and hundreds digits. Which system of equations can be used to find each digit?.
Using the system of equations we get the values of 3 digit number as:
238
Given:
The sum of the digits of a three-digit number is 13.
The tens digit, t, is 1 more than the hundreds digit, h. The units digit, u, is 3 more than the sum of the tens and hundreds digits.
we are asked to determine each digit of the given number:
we frame the equation as:
tens digit (t) = 1+h
hundreds digit (h) = h
units digit (u) = 3+(t+h)
hence,
h + (1+h) + (3+(t+h)) = 13
open the brackets.
h+1+h+3+(t+h)=13
substitute t value from the above mentioned state.
h+1+h+3+((1+h)+h)=13
h+1+h+3+1+h+h=13
arrange the like terms and add.
4h + 5 = 13
4h = 13-5
4h=8
h = 8/4
h = 2
hence we get the hundreds digit as 2.
now substitute it to get t and u value.
t = 1+h
t = 1+2
t = 3
tens digit = 3
u = 3+(t+h)
u = 3+(3+2)
u = 3+5
u = 8
units digit = 8
Hence we get the number as 238 whose sum is 13 when added.
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if you could please help me that would be great.
Find the length of the third. If necessary, round to the nearest tenth 18 12
Answer:
Since it appears that you are talking about a triangle, if the third side that you are looking for is the hypotenuse, then you can use the Pythagorean theorem, a^2 + b^2 = c^2. In this case, the length of the side, if it would be the hypotenuse, would be approximately 21.63.
Step-by-step explanation:
Hope this helped! :D
What is the unit rate for 9 pages in 3 hours? Which answer is correct?
(1): 1 page/3 hours
(2): 3 pages/1 hour
(3): 6 pages/1 hour
(4): 9 pages/3 hours
Answer:
(2) 3 pages in 1 hour
Step-by-step explanation: Simply dividing 9/3 we get, 3/1 by unitary method this is the right answer.
a store sold a case of scented candles for $ 17.85 that had been marked up 110 % what was the original price
The original Price of a case of scented candles was 8.5
What is percentage increase?A percent increase refers to how much a percentage has gone up over time.
In order to reach this number, you would need to find the difference between the original value and the final value, subtracting to find the exact total of the decrease.
The formula for calculating Percentage increase is:
(ΔPrice/Original price) × 100
change in price (new price - original price)/original price) × 100
From the question:
Percentage increase = 110%
New price = 17.85
Original price = X
so it we can substitute in the formula
((17.85 - X)/X) * 100 = 110%
Opening brackets
(17.85*100 - X*100)/ X =110
1785 - 100X = 110X (we balance the equation)
(we collect like terms)
1785 = 210X
to find X we divide both sides by 210
1785/210 = X
8.5
The original Price was $8.5
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A tollbooth collected $900 on a day
in which a total of 1,100 cars and
trucks passed through. Cars pay a
$0.75 toll, and trucks pay a $1.25
toll. How many cars paid the toll
that day?
Using system of linear equations, the number of cars and trucks that parked are 950 and 150 respectively.
System of linear equationThe system of linear equations is the set of two or more linear equations involving the same variables. Here, linear equations can be defined as the equations of the first order, i.e., the highest power of the variable is 1. Linear equations can have one variable, two variables, or three variables. Thus, we can write linear equations with n number of variables.
To solve this problem, we have to write out two equations and solve the unknown variable.
x = carsy = trucksx + y = 1100 ...eq(i)
0.75x + 1.25y = 900 ...eq(ii)
From equ(i)
x = 1100 - y ..eq(iii)
Put eq(iii) into eq(ii)
0.75(1100 - y) + 1.25y = 900
y = 150
Put y = 150 in equation (i)
x + 150 = 1100
x = 1100 - 150
x = 950
x = 950, y = 150
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measurements of variables must be sensitive to be able to measure differences between participants. however, if a measurement was not sensitive, and resulted in everyone scoring high, then this is called:
If a measurement was not sensitive, and resulted in everyone scoring high, then this is called a ceiling effect
Given,
Ceiling effect;-
When a large percentage of study participants get the highest scores on the observable variable, it is said that a ceiling effect has occurred. Discrimination between subjects at the top end of the scale is therefore impossible. For instance, a test could result in, say, 50% of the students receiving a perfect score.
Here,
To be able to measure participant differences, variables must be measured with sufficient sensitivity. However, if a test was insensitive and everyone received good scores, this is known as ceiling effect
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(- 9x³-7x²-3x - 1) − (6x³ + 5x² + 2x + 2)
Find the difference
Answer:
(-15x^3 -12x^2 -5x-3)
Step-by-step explanation:
You can't add/subtract numbers that have different exponent numbers so you have to stick with the ones that have the same number
So then the coefficients would be:
x^3 : -9 - 6 = -15
x^2 : -7 -5 = -12
x: -3 -2 =-5
0: -1 - 2 = -3
let g be a group of order 840, and suppose k is a subgroup of g of order 42. if h is a subgroup of g that contains k as a subgroup, what could the order of h be? explain
The order of H can be 84 , 210 or 420
One of the key theorems in abstract algebra is the Lagrange theorem. According to the Lagrange Theorem , In group theory, for any finite group say G, the order of subgroup H of group G is the divisor of the order of G. The order of the group represents the number of elements
According to the question,
Group G has order 840
Also , By Lagrange's theorem
|K| divides |H| , so order of h should be multiple of 42
|H| = 42k ------(1) (for some k)
And , H is subgroup of G
|G| should divides |H|
=> 42k should divides 840
=> k = 840/42 = 20
Since , H is proper subgroup of G
K < 20
So, either k =2 , 5 or 10
Hence , order of H can be 84 , 210 or 421
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A line with a slope of –14 passes through the point (3,3). What is its equation in point-slope form? Use the specified point in your equation. Write your answer using integers, proper fractions, and improper fractions. Simplify all fractions.
The required equation of the line is given as y = -14x + 45.
What is the equation for a straight line?A straight line can be written in the form of equation as, y = mx + c.
Two straight lines intersect each other only at one point.
When two straight lines are parallel to each other the angle between them is zero.
Given that,
The slope of the line is -14.
And, the point it passes through is (3, 3).
The equation of the line for given data can be written as follows,
(y - 3)/(x - 3) = -14
=> y - 3 = -14(x - 3)
=> y = -14x + 42 + 3
=> y = -14x + 45
Hence, the equation of the line for the given point and slope can be written as y = -14x + 45.
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if a string of length thirteen over {a, b, c, d} is chosen at random, what is the probability that it contains at least one pair of adjacent characters that are the same? (round your answer to the nearest tenth of a percent.)
The total number of possible strings that can be chosen is 4^13, since there are 13 characters and 4 possible choices for each character, b, c, d). Now, we need to determine the number of strings that contain at least one pair of adjacent characters that are the same in this probability.
To do this, we'll consider each letter in the string and calculate the number of possible strings that contain at least one pair of adjacent characters that are the same for that letter. For the first letter in the string, there are 4 strings that contain at least one pair of adjacent characters that are the same (aa, bb, cc, dd).For the second letter in the string, there are 4 strings that contain at least one pair of adjacent characters that are the same (ab, be, cd, dc).For the third letter in the string, there are 4 strings that contain at least one pair of adjacent characters that are the same (ac, bd, ca, db.).For the fourth letter in the string, there are 4 strings that contain at least one pair of adjacent characters that are the same (ad, bc, da, cb).
We can continue this process for each letter
We get probability as 83.3%
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7. towers a lookout tower sits on a network of struts and posts. leslie measured two angles on the tower. if m∠1=7x-7°, m∠2=4x 2°, and m∠3=2x 6°, what is m∠1?
The measure of the ∠1 for the given angles of the triangle are found as 1162/13.
Explain the angle sum property of the triangle?The internal angle of a triangle is just the angle formed by two of its sides. It is often referred to as the triangle's interior angle property. Because of this characteristic, a triangle's interior angles add up to 180 degrees.The given angles are-
∠1 = 7x - 7.
∠2 = 4x + 2
∠3 = 2x + 6
By the angle sum property of the triangle.
∠1 + ∠2 + ∠3 = 180
Thus,
7x - 7 + 4x + 2 + 2x + 6 = 180
Simplifying,
13x = 179
x = 179/13
So, ∠1 becomes-
∠1 = 7x - 7
∠1 = 7(179/13)- 7
∠1 = 1162/13
Thus, the measure of the ∠1 for the given angles of the triangle are found as 1162/13.
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Lexi is rehydrating after an intense workout. She drinks 1/4 of a gallon every 50 minutes, or 5/6 of an hour. How many gallons of water does she drink in one hour?
Lexi drinks 3/10 gallon of water in one hour.
What is ratio?Ratio basically compares quantities, that means it show value of one quantity with respect to other quantity.
If a and b are two values, their ratio will be a:b,
Given that,
Lexi drinks water in 50 minutes = 1/4 gallon
To find water that Lexi drinks in one hour,
Use ratio property,
50 minutes = 1/4 gallon
1 minute = 1/(4 x 50)
60 minutes = 60/(4 x 50)
60 minutes = 3/10 gallon
Lexi drinks 3/10 gallon of water in 1 hour.
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the following data on price ($) and the overall score for 6 stereo headphones that were tested by consumer reports were as follows. brand price ($) score bose 180 72 scullcandy 150 78 koss 96 66 phillips/o'neill 72 54 denon 72 42 jvc 48 24 a. does the t test indicate a significant relationship between price and the overall score?
There is significant relation between price and overall score.
Given that,
Following are details on the cost ($) and final rating for the six stereo headphones that Consumer Reports examined. ($) brand price Score: Bose 180, 72 Scullcandy 150, 78 Koss, 96, 54 Phillips/O'Neill, 72, 42 Denon, 48, 24.
a) t =3.54
The P value is less than 0.2
There is significant relation between price and overall score.
P value is between 0.01 and 0.025.
Because p value is less than or equal to 0.05 , we reject H₀ : β₁ is equal to zero.
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Software companies work hard to produce software without bugs. A particular company claims that 85% of the software it produces is bug free. A random sample of size 200 showed 156 softwareprograms were bug free.
a. Calculate the mean of the sampling distribution of the sample proportion.
the sample proportion under probability has a mean sampling distribution of 82.
The population percentage is the major variable in the sampling distribution of this proportion, which has a quarter-um normal distribution. Variants include population proportion times one and population proportion divided by sample size. Additionally, we are aware that the population proportion is under the null hypothesis. 200 is the sample soil. When we enter the value of P into this calculation without normalisation, we can obtain the standing ovation as the variance's root. 025 in excess. The probability that we got a sample result of 188/58 or less is then calculated. We simply need to determine this left tail probability after that.
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all of the following increase the width of a confidence interval except a. decreased sample size b. increased sample size c. increased confidence level d. increased variation
The width of a confidence interval except increased sample size.
What is the Confidence Interval?
An area surrounding a measurement that indicates how accurate it is is called a confidence interval.
The following points clear the answer of given question:
The width of the confidence interval decreases as the sample size increases.The width increases as the standard deviation increases.The width increases as the confidence level increases. The width increases as the significance level decreases.Hence, The width of a confidence interval except increased sample size.
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On a recent test, students had to determine whether the relation represented by the ordered pairs (4,5), (9,15), (14,22), (18,37) is a function. Bobby drew the arrow diagram on the right and said the relation was not a function. What error did Bobby most likely make? (The answer is NOT B I just chose that before and it was wrong!)
The error that Bobby most likely made in verifying if the relation is a function is given as follows:
C. Bobby drew the incorrect arrow diagram. The correct arrow diagram of the relation is a function. Bobby's arrow diagram is not a function.
When does a relation represents a function?A relation represents a function if each value of the input is mapped to only one value of the output.
The ordered pairs from this data-set are given as follows:
(4,5), (9,15), (14,22), and (18,37).
The input and the output for the pairs are given as follows:
The input of 4 is mapped to the output of 5.The input of 9 is mapped to the output of 15.The input of 14 is mapped to the output of 22.The input of 18 is mapped to the output of 37.Each value of the input is mapped to only one value of output, hence the relation represents a function.
However, Bobby made a mistake, in mapping the point (9, 15), as he mapped an input of 14 to an output of 9, and thus it would not represent a function. Hence the correct statement is given by option C.
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Write the equation of the circle centered at (5,-2) with diameter 14.
Answer:
Step-by-step explanation:
550 tickets were sold for a game for a total of $687.50. if adult tickets sold for $1.50 and children's tickets sold for $1.00, how many of each kind of ticket were sold?
9514 1404 393 each kind of ticket were sold if 550 tickets were sold for a game for a total of $687.50. if adult tickets sold for $1.50 and children's tickets sold for $1.00,
adult: 325
children's: 225
It usually works well to let a variable represent the higher-value item in the mix. Here, we can let 'a' represent the number of adult tickets sold. Then the total revenue is ...
1.50a +1.00(550 -a) = 687.50
0.50a = 162.50 . . . . . . . . . . . . . subtract 550 and collect terms
a = 325
c = 550 -325 = 225
325 adult and 225 children's tickets were sold
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the correlation between two variables related to the same sample is a scaled quantitative measure. what is used to scale this measurement?
The correct option is the standard deviations of the variables.
What is correlation?
Any statistical relationship between two random variables or bivariate data, whether causal or not, is referred to in statistics as correlation or dependence.
Correlation is a term used to describe the linear relationship between two entities. When using a correlation tool, the variables must be present in quantities of two or more to test their relationship (degree of association).
In order to scale the measurement between two variables connected to a similar sample, the standard deviation measures dispersion.
The variables' standard deviations are the best choice as a result.
Option involving the standard errors of the variables is incorrect because standard error, which stands for standard deviation, determines the relationship between dependent and independent variables.
Option involving the coefficients of variation of two variables is incorrect because the coefficient expressing standard deviation and mean ratio determines how much variation there is.
Option The means of the variables are incorrect because they are not a scale for determining the correlation between two variables.
Hence, the correct option is the standard deviations of the variables.
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Complete question:
The correlation between two variables related to the same sample is a scaled quantitative measure. What is used to scale this measurement?
- the standard deviations of the variables
- the standard errors of the variables
- the coefficients of variation of the two variables
- the means of the variables
Suppose it takes 4 hours for a certain strain of bacteria to reproduce by dividing in half. If 45 bacteria are present o begin with the total number present after a days is f(x) = 45 - 64* Find the total number present after 1, 2 and 3 days. There will be bacteria present after 1 day, after 2 days and after 3 days.
A certain bacteria strain needs 4 hours to double its population. If in the beginning there are 45 bacteria, after 1 day, the number of bacteria present is 2,880, after 2 days, the number of bacteria is 184,320, after 3 days, the number of bacteria is 11,796,480.
The problem can be solved using the exponential growth model.
In the given problem, the formula for the growth model is given, that is:
f(x) = 45 (64)^(x)
Where:
f(x) = total number of bacteria present after x days
To find f(x) for 1 day, 2 days, and 3 says, substitute x = 1, x = 2, and x = 3 into the formula.
f(1) = 45 (64)¹ = 2,880
f(2) = 45 (64)² = 184,320
f(3) = 45 (64)³ = 11,796,480
Hence,
after 1 day, the number of bacteria present is 2,880, after 2 days, the number of bacteria is 184,320, after 3 days, the number of bacteria is 11,796,480.
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A hobby store prices model train track using a proportional relationship between the length of track (in inches) and the cost in dollars. If 6. 4 inches of track costs $16, what is the constant of proportionality?.
$2.50 is found to be the proportionality constant for the relationship between the length of track (in inches) and the cost in dollars.
Define the term constant of proportionality?The ratio connecting two given numbers in what is regarded as a proportional relationship is the constant of proportionality. Constant ratio, consistency index, unitary rate, constant variable variation, and even rate of change are other names for said constant of proportionality.Direct proportionality is represented by the equation y = kx.
Where:
dependent variable = yx = unrelated variableK is the proportionality constant.The price of the track is the dependent variable in this situation.
The tracks' length is an independent variable.
$16 = 6.4k
k = 16 / 6.4
k = $2.5
Thus, the constant of proportionality for the relationship between the length of track (in inches) and the cost in dollars is found as $2.5.
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=
Find the quadratic function y = a(x-h)2 whose graph passes through the given points.
(12,-10) and (8,0)
Answer:
y = -5/8(x -8)²
Step-by-step explanation:
You want the function of the form y = a(x -h)² that passes through the points (12, -10) and (8, 0).
Vertex formThe vertex form of the equation of a quadratic is ...
y = a(x -h)² +k . . . . . . quadratic with vertex (h, k)
Comparing this to the given equation, we see that k=0. This means the point (8, 0) is the vertex and h = 8.
Scale factorThe value of 'a' can be found using the other point.
(x, y) = (12, -10)
-10 = a(12 -8)²
-10 = 16a
-5/8 = a . . . . . . divide by 16 and reduce the fraction
The quadratic function is y = -5/8(x -8)².
__
Additional comment
The equation can also be found using technology to do a quadratic regression with the constraint that the vertex is on the x-axis.
What’s higher -12 or -24
What kind of triangle is this?
Answer:
Obtuse
Step-by-step explanation:
Obtuse because one angle is more than 90 degrees.
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suppose there are n independent trials of an experiment with k>3 mutually exclusive outcomes, where p i represents the probability of observing the ith outcome. what would be the formula of an expected count in this situation?
The probability for observing the expected count will be given by [tex]E_i = nP_i[/tex]
Consider an experiment with n independent trials and k > 3 mutually incompatible outcomes.
Where [tex]P_i[/tex] stands for the likelihood of seeing the i-th result.
Hence for the nth expected result on i will be given by multiplying the i-th probability with the expected variable n.
Hence [tex]E_i = n\times P_i[/tex]
∴ [tex]E_i = nP_i[/tex] provides the predicted numbers for each potential outcome.
where pi is the likelihood of seeing the i-th result.
The predicted count of the i-th result is called [tex]E_i[/tex].
When two events cannot occur concurrently at the same moment, they are said to be mutually exclusive in accordance with probability theory. Or, to put it another way, discontinuous events are said to be mutually exclusive.
There is no possibility of two occurrences occurring simultaneously if they are viewed as separate events. If two events do not take place at the same time, they are said to be mutually exclusive. If a coin is tossed, the outcome will always be either head or tail; we cannot get both.
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you have been asked to assemble a two person team from among your five person team. how many different teams are there
If you have been asked to assemble a two person team from among your five person team, then there are 10 different teams
Total number of members for the team = 5
Number of members that we need to assemble = 2
Here we have to use the combination
The combination is defined as the number of ways of selecting samples from the collection and the order of the collection does not matter
The number of combination of 2 members from a 5 member team = [tex]5C_2[/tex]
= 5! / 2!(5-2)!
= 5! / 2! × 3!
= (5 × 4 × 3!) /(2! × 3!)
= (5 × 4) / 2
= 20 / 2
= 10
Therefore, there are 10 different teams
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