A six-person committee composed of Alice, Ben, Connie, Dolph, Egbert, and Francisco is to select a...

A six-person committee composed of Alice, Ben, Connie, Dolph, Egbert, and Francisco is to select a chairperson, secretary, and treasurer. Nobody can hold more than one of these positions.

a. How many selections are there in which Dolph is either a chairperson or he is not an officer?

b. How many selections are there in which Ben is either chairperson or treasurer?

c. How many selections are there in which either Ben is chairperson or Alice is secretary?

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Expert Solution

Answer:

Given that:

A six-person committee composed of Alice, Ben, Connie, Dolph, Egbert, and Francisco is to select a chairperson, secretary, and treasurer. Nobody can hold more than one of these positions.

(a) How many selections are there in which Dolph is either a chairperson or he is not an officer?

cases in which Dolph is either a chairperson or he is not an officer are independent cases.

cases in which Dolph can be chairperson will be 5P2 = 20

cases in which Doplh is not an officer will be 5P3 = 60

hence total number of cases will be =80

(b) How many selections are there in which Ben is either chairperson or treasurer?

total number of cases in which Ben is either a chairperson or treasurer will be ⁵P₂ + ⁵P₂

= 20 + 20 = 40 because both of these events can't hold simultaneously

(c) How many selections are there in which either Ben is chairperson or Alice is secretary?

cases in which either ben is a chairperson or alice is secretary will be ⁵P₂ + ⁵P₂ - ³P₁ = 20 + 20 – 3 = 37 . Here (-3) because there are possibility that ben is chairperson and alice is secretary simultaneously.

orchestra answered 1 year ago

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