In how many ways can a committee consisting of 4 faculty members and 5 students be formed if there are 7 faculty members and 11 students eligible to serve on the committee?

 16170 




Work Shown

We have 7 faculty members and we're selecting 4 of them to be placed on the committee. Since order doesn't matter, we use the nCr combination formula

In this case, n = 7 and r = 4
n C r = (n!)/(r!*(n-r)!)
7 C 4 = (7!)/(4!*(7-4)!)
7 C 4 = (7!)/(4!*3!)
7 C 4 = (7*6*5*4!)/(4!*3!)
7 C 4 = (7*6*5)/(3!)
7 C 4 = (7*6*5)/(3*2*1)
7 C 4 = 210/6
7 C 4 = 35
There are 35 ways to pick the four faculty members (from a pool of seven). Order does not matter.
We'll use this value later. Call this value M. So M = 35.



We also are selecting 5 students from a pool of 11. Order doesn't matter so we can use nCr again. This time n = 11 and r = 5.
n C r = (n!)/(r!*(n-r)!)
11 C 5 = (11!)/(5!*(11-5)!)
11 C 5 = (11!)/(5!*6!)
11 C 5 = (11*10*9*8*7*6!)/(5!*6!)
11 C 5 = (11*10*9*8*7)/(5!)
11 C 5 = (11*10*9*8*7)/(5*4*3*2*1)
11 C 5 = 55440/120
11 C 5 = 462
There are 462 ways to pick the five faculty members (from a pool of eleven). Order does not matter.
We'll use this value later. Call this value N. So N = 462



Now we'll use the values of M and N. Specifically we multiply them to get the final answer

M*N = 35*462 = 16170

There are 16170 ways to form the entire committee (four faculty members; five students)