Homework SoursePart 1 Suppose you want to line up people for a photograph T
Part 1: Suppose you want to line up people for a photograph. There are 7 women, 8 men and 5 children.
(a) In how many ways can you arrange all 20 people?
(b) In how many ways can you arrange all 20 people if the women all appear consecutively, the men all appear consecutively and the children all appear consecutively?
(c) In how many ways can you arrange all 20 people if the children all appear consecutively?
(d) In how many ways can you arrange all 20 people if the first man appears before the first child? (You can assume the positions are ordered from left to right.)
(e) In how many ways can you arrange 9 out of the 20 people? (f) In how many ways can you arrange 9 out of the 20 people if all 5 children are included?
(g) In how many ways can you arrange 9 out of the 20 people if all 5 children are included, at least one women is included and at least one man is included.
Part 2: You form a team of six from 20 players. (Terry, Sam, Dean and Mark are four of these 20.) In how many ways can this be done
(a) with no restrictions?
(b) if Terry and Sam must be on the team?
(c) no two of Sam, Dean and Mark can be on the team?
(d) if Terry must be on the team, and at most two of Sam, Dean and Mark can be on the team?
Note: It is all one question as written in UPEI combinatorics midterm fall 2016
Solution
part 1
a) In how many ways can you arrange all 20 people?
Ans-number of ways 20 people canb be arranged in a row , is 20! ( 20 factorial)
b)In how many ways can you arrange all 20 people if the women all appear consecutively, the men all appear consecutively and the children all appear consecutively?
Answer- (7!8!5!)*3!
(c) In how many ways can you arrange all 20 people if the children all appear consecutively?
Answer- (15!5!)*2!
(e) In how many ways can you arrange 9 out of the 20 people?
Answer - 20C9
(f) In how many ways can you arrange 9 out of the 20 people if all 5 children are included?
Answer = 5C5*15C4
(g) In how many ways can you arrange 9 out of the 20 people if all 5 children are included, at least one women is included and at least one man is included.
G)5c5*7c1*8c3+5c5*7 c2*8c2+5c5*7c3*8c1
