Question

A class committee is to be made that consists of 2 freshmen, 2 sophomores, 3 juniors, and 3 seniors. If students are being randomly chosen from groups of 12 freshmen, 17 sophomores, 16 juniors, and 20 seniors, how many possible committees can be formed?

Answer 1

Freshmen: ₁₂C₂ $\frac{12!}{2!(12 - 2)!} = \frac{12!}{2!*10!} = \frac{12 * 11 * 10!}{2*10!} = \frac{12 * 11}{2} = 6 * 11 = 66$

Sophomore: ₁₇C₂  = $\frac{17!}{2!(17 - 2)!} = \frac{17!}{2!*15!} = \frac{17 * 16 * 15!}{2 * 15!} = \frac{17 * 16}{2} = 17 * 8 = 136$

Juniors: ₁₆C₃ = $\frac{16!}{3!(16 - 3)!} = \frac{16!}{3!*13!} = \frac{16 * 15 * 14 * 13!}{3 * 2 * 13!} = \frac{16 * 15 * 14}{2 * 3} = 8 * 5 * 14 = 560$

Seniors: ₂₀C₃ = $\frac{20!}{3!(20 - 3)!} = \frac{20!}{3!*17!} = \frac{20 * 19 * 18 * 17!}{3*2*17!} = \frac{20 * 19 * 18}{2 * 3} = 10 * 19 * 6 = 1140$

₁₂C₂ * ₁₇C₂  * ₁₆C₃  * ₂₀C₃

66  * 136 *  560 * 1140 = 5,730,278,400

Answer: 5,730,278,400

Answer 2

$C(12,2)\cdotC(17,2)\cdotC(16,3)\cdotC(20,3)=\\\\\dfrac{12!}{2!10!}\cdot\dfrac{17!}{2!15!}\cdot\dfrac{16!}{3!13!}\cdot\dfrac{20!}{3!17!}=\\\\\dfrac{11\cdot12}{2}\cdot\dfrac{16\cdot17}{2}\cdot\dfrac{14\cdot15\cdot16}{2\cdot3}+\dfrac{18\cdot19\cdot20}{2\cdot3}=\\\\66\cdot136\cdot560\cdot1140=5,730,278,400$