Question

The Mathletes Club has 15 members and is electing a 5-member committee (president, vice-president, secretary, treasurer, and sergeant-at-arms). One of the members, Alice, has set a condition that, if she is not elected president, then she will not accept any position in the committee. In how many ways can the club select the committee?

Answer 1

The total number of ways are 264,264

Explanation:

Consider the provided information.

The Mathletes Club has 15 members and is electing a 5-member committee.

One of the members, Alice, has set a condition that, if she is not elected president, then she will not accept any position in the committee.

Case I: If she is elected as president.

If she is elected as president so now we have 14 members and is electing a 4-member committee. (vice-president, secretary, treasurer, and sergeant-at-arms).

No ways:
`^(14)C_4* 4!=1001* 24=24024`

Case II: If she is not elected as president.

If she is not elected as president so now we have 14 members and is electing a 5-member committee. (president, vice-president, secretary, treasurer, and sergeant-at-arms)

No ways:
`^(14)C_5* 5!=2002* 120=240240`

Hence, the total number of ways are:

24,024+240,240=264,264

Hence, the total number of ways are 264,264