Question

There are members on the board of directors for a certain​ non-profit institution. a. If they must elect a​ chairperson, first vice​ chairperson, second vice​ chairperson, and​ secretary, how many different slates of candidates are​ possible? b. If they must form an ethics subcommittee of four​ members, how many different subcommittees are​ possible?

Answer 1

Answer: a. 1320

b.495

Step-by-step explanation:

Complete question is provided in the attachment.

Total members on the board = 12

a. Persons to chose : chairperson, first vice​ chairperson, second vice​ chairperson, and​ secretary

i.e. Total 3 posts in an order.

Number of ways to choose 3 persons from 12 in an order = $^{12}P_3$  [By permutation]

$=\dfrac{12!}{(12-3)!}\\\\=\dfrac{12!}{7!}\\\\=12\times11\times10\\\\=1320$

hence, 1320 different slates of candidates are​ possible .

b. number of ways to choose 4 members out of 12 ( order not matters)=$^{12}C_{4}$ [By combinations]

$=\dfrac{12!}{4!8!}\\\\=\dfrac{12\times11\times10\times9}{24}\\\\=495$

Hence, the number of different subcommittees are​ possible =495