Question 1191897: A Student Community needs to select 4 vice presidents and 2 secretaries among 10 male students and 5 female students. How many ways you can create this committee with 3 male vice presidents 1 female vice president and secretaries from both genders? (1 male secretary and 1 female secretary)
Answer by math_tutor2020(3816)   (Show Source):
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There are 10 male students and we need to select 3 of them to get the male VPs
Use n = 10 and r = 3 with the nCr combination formula
Order doesn't matter.
n C r = (n!)/(r!(n-r)!)
10 C 3 = (10!)/(3!*(10-3)!)
10 C 3 = (10!)/(3!*7!)
10 C 3 = (10*9*8*7!)/(3!*7!)
10 C 3 = (10*9*8)/(3!)
10 C 3 = (10*9*8)/(3*2*1)
10 C 3 = (720)/(6)
10 C 3 = 120
Alternatively, you can use Pascal's Triangle to look at the row that starts with 1,10,... to locate the fourth item which corresponds to 10C3 = 120.
There are 120 ways to pick the 3 male VPs where order doesn't matter.
There are 5 ways to pick the 1 female VP since we have 5 female students.
There are 120*5 = 600 ways to pick the VPs.
After the VPs are accounted for, we have 10-3 = 7 males left and 5-1 = 4 females left.
There are 7*4 = 28 ways to pick 1 male secretary and 1 female secretary if order mattered. However, order doesn't matter so we divide by 2 to get 28/2 = 14.
We found that,
There are 600 ways to pick the VPs
There are 14 ways to pick the secretaries
Therefore, we have 600*14 = 8400 ways to form the committee.
Answer: 8400
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