Question

Thirty students are members of the Biology Club: 21 are men (one of whom is named Joe) and 9 are women (one of whom is named Kate). Five students are to be selected (unordered, without replacement) to form the Leadership Committee.

12a) How many different Leadership Committees are possible?

12b) Refer to the previous question. How many options are there if Joe can’t be on the committee? How many options are there if Joe must be on the committee? Hint: should these two values add up to your previous answer?

12c) How many different Leadership Committees are possible if it must consist of exactly 3 men and 2 women?

12d) How many different Leadership Committees are possible if Joe and Kate must be on the committee? Hint: notice there are no requirements on whether or not the other three members are men or women.

Answer 1

a) number of ways to select 5 student from 30 =(₃₀C₅) =142506

b) options are there if Joe can’t be on the committee =select 5 from remaining 29=(₂₉C₅) =118755

options are there if Joe must be on the committee =select 4 from remaining 29=(₂₉C₄) =23751

c)

number of ways =(₂₁C₃)*(₉C₂) =1330*36 =47880

d)

number of ways =N(select remaining 3 from remaining 28) =(₂₈C₃)=3276