If the team captain wants to be at either end of the line, how many ways can the team members line up?

Question

Can someone just check my answers and help me with part b please?

The volleyball team is lining up to take a yearbook photo. There are 12 team members, three of which are seniors.

a. How many ways can the team members line up for the photo?

$12!$

b. If the team captain wants to be at either end of the line, how many ways can the team members line up?

This one I'm confused about because I don't know if I'm permuting two spaces or one. Because it doesn't matter if he is at the front or the back so would it be

$11!(\frac{2!}{2!})$ or would it be 11P2

c. If the three seniors want to stand next to each other, how many ways can the team members line up?

$11!(\frac{3!}{3!})=11!$

Answer 1

When captain is at start or end of row then there are 2×11! ways . All you did is just that you fixed captain's position from 12 places to two . $(12! = 12×11!)$

For 3rd part Assume all three seniors as one so now there are 10 persons to rearrange so there are total $10!$ ways . And if you take all seniors different then you can also rearrange them , so there will be $10!×3!$ ways .

Answer 2

HINT:

For part (b), apart from the captain, the $11$ other members can stand in $11!$ ways. For each of these $11!$ permutations, the captain can take two positions (either end of the line). So what is the total number of posibilities?

For part (c), $9$ junior members + the set of seniors can stand in $(9+1)!$ ways. Also, among themselves, the seniors can stand in $3!$ ways. So how many ways can the members line up?