There are nine seniors and seven juniors in the Math Club at

There are nine seniors and seven juniors in the Math Club at Jefferson High School. In how many ways can a math team consisting of four seniors and two juniors be selected from members of the Math Club?

Solution

So there are total 9 seniors and 7 junior members.

so we need to select 4 seniors form total number of 9 seniors then

total number of ways is C(9,4)

and we have 7 junior members and we have to select 2 juniors then

total number of ways is C(7,2)

so total possible ways is C(9,4) * C(7,2)

Where C(a,b) = a! / ((a-b)! * b!)

so C(9,4) = 9! / ((9-4)! * 4!) = 9! / ((5)! * 4!)

= (9 x 8 x 7 x 6 x (5! )) / ((5)! * 4!)

= (9 x 8 x 7 x 6 ) / (4 x 3 x 2 x 1)

  C(9,4) = 126

and C(7,2) =  7! / ((7-2)! * 2!) = 7! / (5)! * 2!)

= (7 x 6 x 5!) / ( (5)! * 2!))

= (7 x 6 )/ (2 x 1) = 21

C(7,2) = 21

so total possible ways = 126 x 21 = 2646