There are 40 people on the faculty. Of the 40, 15 are men and 25 are women. Use
ID: 3258752 • Letter: T
Question
There are 40 people on the faculty. Of the 40, 15 are men and 25 are women. Use this information to answer the following questions:
a) A committee of 8 is to be selected by the principal. How many possible committees can be created, regardless of gender?
b) i. all men
ii. all women
iii. 4 men and 4 women
iv. at least 2 men
v. no more than 3 men
iv. 5 women and 3 men
c) The principal now has his 8 members. Answer the following questions about these 8. Assume only 8 chairs for them to sit.
i. if they sit at large table (in a line), jow many ways can they be seated?
ii. how many ways can they be seated in a circle?
iii. if they are in a line, how many ways can they be seated if 3 people have to sit together?
iv. i am on the committee and can't stand 2 of the members. how many seating arrangements can be made if I cannot sit next to either of these 2?
Explanation / Answer
Given
There are 40 people on the faculty.
Of the 40, 15 are men and 25 are women
a) A committee of 8 is to be selected by the principal. How many possible committees can be created, regardless of gender?
No of ways that a committe of 8 members can form is we have to select 8 members from 40 members given that regardless of gender = 40C8
b) i). all men
we have to select all 8 men for the 8 men committe
total members present = 15
of these we have to select 8 members
No of ways = 15C8
ii) all women
toatl no of women = 25
total no. of ways = 25C8
iii)4 men and 4 women
there are 25 women and 15 men
no of ways = 25C4 * 15C4
iv) at least 2 men
atleast 2 men means selecting 2 or more men i.e(2,3,4,5,6,7,8)
no . of ways = 15C2 *25C6 + 15C3 * 25C3 +15C4 * 25C4 + 15C5 *25C3 + 15C6 * 25C2 + 15C7 * 25C1 + 15C8
v) no more than 3 men
No more than 3 men means less than 3 men to be selected i.e( 0 , 1 , 2 , 3 )
no. of ways = 25C8 + 25C7 * 15C1 + 25C6 * 15C2 + 25C5 * 15C3
vi). 5 women and 3 men
No . of ways = 25C5 * 15C 3
C) The principal now has his 8 members. Answer the following questions about these 8. Assume only 8 chairs for them to sit.
i. if they sit at large table (in a line), How many ways can they be seated?
8 members are selected we have to arrange the in 8 seats = 8! ways
ii. how many ways can they be seated in a circle?
8 members are to be seated in 8 seats in circle = (8-1)!
= 7!
iii)if they are in a line, how many ways can they be seated if 3 people have to sit together?
the group of 3 people can be taken as 1
then total no . of ways = 5 people + 1 group
= 6! ways
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