Question 1200584: Each of the teams in a mixed volleyball league consists of 5 men and 5 women. There are 9 men and 10 women, including Julia, who are trying out for a team. How many possible teams would include Julia?
15,876
31,752
45,722,880
457,228,800
Found 2 solutions by ikleyn, math_tutor2020:
Answer by ikleyn(52778)   (Show Source):
Answer by math_tutor2020(3816)  (Show Source):
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There are 9 men and 5 spots for them.
Order does not matter.
n = 9
r = 5
We use the nCr combination formula.
n C r = (n!)/(r!(n-r)!)
9 C 5 = (9!)/(5!*(9-5)!)
9 C 5 = (9!)/(5!*4!)
9 C 5 = (9*8*7*6*5!)/(5!*4!)
9 C 5 = (9*8*7*6)/(4!)
9 C 5 = (9*8*7*6)/(4*3*2*1)
9 C 5 = (3024)/(24)
9 C 5 = 126
There are 126 ways to select the five men from a pool of nine male candidates.
Because Julia is on the team, there are 10-1 = 9 women left to pick from and 5-1 = 4 female roster spots.
n = 9
r = 4
n C r = (n!)/(r!(n-r)!)
9 C 4 = (9!)/(4!*(9-4)!)
9 C 4 = (9!)/(4!*5!)
9 C 4 = (9*8*7*6*5!)/(4!*5!)
9 C 4 = (9*8*7*6)/(4!)
9 C 4 = (9*8*7*6)/(4*3*2*1)
9 C 4 = (3024)/(24)
9 C 4 = 126
There are 126 ways to select the remaining four female roster spots from a pool of nine remaining female candidates.
Side note: You can use Pascal's Triangle to determine the result of 126 calculated earlier. Look at the row that has 1,9,... and look at the 5th and 6th slots.
We found that there are...- 126 ways to pick the men
- 126 ways to pick the women, where Julia is on the team.
That will give 126*126 = 15876 different teams where Julia is guaranteed to be on it.
Here is a calculator that computes nCr values
https://www.calculatorsoup.com/calculators/discretemathematics/combinations.php
Answer: 15876 (choice A)
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