QUESTION 10
20
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?
Answers
Answer:
Freshmen: ₁₂C₂ \frac{12!}{2!(12 - 2)!} = \frac{12!}{2!*10!} = \frac{12 * 11 * 10!}{2*10!} = \frac{12 * 11}{2} = 6 * 11 = 66
2!(12−2)!
12!
=
2!∗10!
12!
=
2∗10!
12∗11∗10!
=
2
12∗11
=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
2!(17−2)!
17!
=
2!∗15!
17!
=
2∗15!
17∗16∗15!
=
2
17∗16
=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
3!(16−3)!
16!
=
3!∗13!
16!
=
3∗2∗13!
16∗15∗14∗13!
=
2∗3
16∗15∗14
=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
3!(20−3)!
20!
=
3!∗17!
20!
=
3∗2∗17!
20∗19∗18∗17!
=
2∗3
20∗19∗18
=10∗19∗6=1140
₁₂C₂ * ₁₇C₂ * ₁₆C₃ * ₂₀C₃
66 * 136 * 560 * 1140 = 5,730,278,400
Answer: 5,730,278,400