Question

A 6 person committee is to select a chairperson, secretary, and treasurer. Nobody can hold more than one of these positions.

a. Connie and Dolph are both members of the committee. How many selections exclude both Connie and Dolph?

b. Dolph is a member of the committee. How many selections are there in which Dolph is either a chairperson or he is not an officer?

c. Ben and Alice are members of the committee. How many selections are there in which either Ben is chairperson or Alice is secretary?

Answer 1

total number of person in committee ( n ) = 6

a.

after exclude both Connie and Dolph ( m ) = 6 - 2

3 person selection as chairperson, secretary, and treasure from m is = mC3 $\times$ 3! = 4C3 $\times$ 3! = 24

where mC3 is selection of 3 person and 3! is give position to selected person as chairperson, secretary, and treasure.

b.

total number of selection = if Dolph is  chairperson +  he is not an officer

if Dolph is chairperson then = 5C2 $\times$ 2! = 20 ( select secretary, and treasure after excluding Dolph because Dolph is already Chairperson)

if he is not an officer then =  5C3 $\times$ 3! = 60 ( select  chairperson, secretary, and treasure after excluding Dolph because Dolph is not officer)

Dolph is either a chairperson or he is not an officer = 20 + 60 = 80

c.

Let, A event of Alice is secretary and B event for Ben is chairperson

number of selection when Ben is chairperson or Alice is secretary is = n(A$\tiny \bigcup$B)

n(A $\tiny \bigcup$ B) = n(A) + n(B) - n( A $\tiny \bigcap$ B)

where n (A ) is number of selection when Alice is secretary = 5C2 $\times$ 2! = 20

n(B) is number of selection when Ben is chairperson =  5C2 $\times$ 2! = 20

n( A $\tiny \bigcap$ B) is number of selection when Alice is secretary and Ben is chairperson = 4C1 = 4

Now   n(A $\tiny \bigcup$ B) = 20 +20 - 4

= 36

number of selection when Alice is secretary or Ben is chairperson 36 .