You are playing billiards There are 15 balls on the table sa

You are playing billiards. There are 15 balls on the table (save the cue ball which we are ignoring) numered 1, 2, ... , 15 and 6 pockets the balls can go into (4 corner pockets and two side pockets, displayed below). The goal of the game you are playing is to get all the 15 balls into any of the pockets.

a)How many ways to sink all balls if pockets can be empty and both balls and holes are indistinct?In detail.

b)How many ways to sink all balls if pockets can be empty with the restriction that each pocket must have at least one odd ball and at least one even ball?In detail.

Solution

There are 15 balls and 6 pockets

a) Balls as well as holes are identical.

Also any pocket can be empty.

Each ball has 6 choices of pockets.

Hence total no of ways to sink all balls = 15⁶

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b) In this pockets can be empty

Also each pocket should have one odd and one even.

No of ways for I ball = 6

II ball