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

1. You are playing billiards. There are 15 balls on the table (save the cue ball which we are ignoring) numbered 1, 2, ..., 15 and 6 pockets the balls can go into (4 corner pockets and two side pockets) 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 each pocket must have at least one odd ball and at least one even ball?

b) How many ways to sink all balls if pockets can be empty and both balls and holes are indistinct?

Please show all work and explain in full detail. Thanks!!

Solution

Total balls 15

Total pockets = 6

a) The restrisction here is that each pocket must have atleast one odd and one even ball

Total odd balls = 1,3,5,7,9,11,13,15 -- 8

Even balls = 7

First pocket can have any of 8 odd balls and 7 even balls

No of ways for first pocket = 8x7

II pockets can have any of remaining 7 odd x6 even

III pocket any one of 6 odd x5 even

IV                          5        x4

v                            4        x3

vi                           3        x2

Now left over balls are 2 odd and 1 even

These can go to any pocket hence 3^6 ways

Total no of ways ={ (8x7)(7x6)...(3x2)}3⁶

=

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b) POckets can be empty and balls and holes are indistinct

First ball can go to any of the 6 pockets, ii 6 pockets , etc

15⁶ ways