A six person committee composed of Alice, Ben, Connie, Dolph, Egbert, and
Francisco is to select a chairperson, secretary, and treasurer.
(a) In how many ways can this be done?
(b) In how many ways can this be done if either Alice or Ben must be chairperson?
(c) In how many ways can this be done if Egbert must hold one of the offices?
(d) In how many ways can this be done if both Dolph and Francisce must hold
office?

a.) 20 ways

b.) 12 ways

c.) 10 ways

d.) 4 ways

Step-by-step explanation:

This question deals with selection of a few from a general larger possible options and the combination formula for selection is used to solve this question. This combination formula is denoted by:

nCr = n!/(n-r)! × r!

a.) to select the people to fill the 3 positions from the possible 6 options, Number of ways become: 6C3 = 6!/(6-3)! × 3! = 20Ways

b.) If Alice or Ben must be the chair person, then we have two possibilities.

If Alice is Chairperson, then we chose other 2 positions from the remaining 4 members or Ben be the chairperson and we choose the remaining 2 positions from the 4 members left. Number of ways of this becomes:

number of ways to choose chairperson × number of ways to choose other two posts = 2C1 × 4C2 = 12ways.

c.) If Egbert must hold one of the offices, then we have just 2 positions left to be chosen from 5 member. Number of ways for this = 5C2 = 10ways

d.) If Dolph and Francisce must hold office then the last office is to be given to Just 1 among the 4 possible members. Number of ways for this = 4C1 = 4ways.

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step-by-step explanation:

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