Five distinct boxes labeled 1,2,...,5 contain distinct objects. If box 1 contain

Question

Five distinct boxes labeled 1,2,...,5 contain distinct objects. If box 1 contains 1 object, box 2 contains 2 objects, and in general, box j contains j objects. We are allowed to choose only one object from any box (Order does not matter in the selection for the questions below).

(a) how many ways are there to choose 2 objects from separate boxes?

(b) how many ways are there to choose 3 objects from different boxes?

(c) how many ways are there to choose 5 objects from different boxes?

Please show work and explain how you got the answers for each. I would really appreciate it. Thank you in advance.

Explanation / Answer

(a) number of ways to choose 2 objects from separate boxes:

= 5C2

= 5! / (2! * 3!)

= 10

(b) number of ways to choose 3 objects from different boxes:

= 5C3

= 5! / (3! * 2!)

= 10

(c) number of ways to choose 5 objects from separate boxes:

= 5C5

= 5! / (5! * 0!)

= 1

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