math discrete

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(a)Sarah wants to make a bracelet. She needs some beads and goes to
a shop where they sell beads in exactly 100 different colours.

i. How many ways can she pick 25 beads, each of a different colour?
ii. How many ways can she pick any 25 beads?

(b) Suppose that Sarah has picked 25 beads each of different colours.

i If Sarah arranges all of these beads in a line on the table in
front of her how many different arrangements are possible? What
about if she arranges them in a circle on the table, how many
arrangements are possible if two arrangements are considered
identical if they can be obtained from one another by rotation?

ii. If Sarah uses all of the 25 beads to make her bracelet, how many
distinct bracelets can she make ?

iii. What if she decides to make her bracelet using at least 20 of the
beads, how many distinct bracelets can she make then?

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