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

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

Answer 2

Freshmen: ₁₂C₂
`(12!)/(2!(12 - 2)!) = (12!)/(2!*10!) = (12 * 11 * 10!)/(2*10!) = (12 * 11)/(2) = 6 * 11 = 66`

Sophomore: ₁₇C₂ =
`(17!)/(2!(17 - 2)!) = (17!)/(2!*15!) = (17 * 16 * 15!)/(2 * 15!) = (17 * 16)/(2)  = 17 * 8 = 136`

Juniors: ₁₆C₃ =
`(16!)/(3!(16 - 3)!) = (16!)/(3!*13!) = (16 * 15 * 14 * 13!)/(3 * 2 * 13!) = (16 * 15 * 14)/(2 * 3) = 8 * 5 * 14 = 560`

Seniors: ₂₀C₃ =
`(20!)/(3!(20 - 3)!) = (20!)/(3!*17!) = (20 * 19 * 18 * 17!)/(3*2*17!) = (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