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Question 29 : In how many ways can 8 identical pens be distributed among Amal, Bimal, and Kamal so that Amal gets at least 1 pen, Bimal gets at least 2 pens, and Kamal gets at least 3 pens? [TITA]

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Let us assume Amal gets 1 + x pens and Bimal gets 2 + y pens and Kamal gets 3 + z pens

1 + x + 2 + y + 3 + z = 8

x + y + z = 2 (x , y , z being whole numbers) or

x + (1 + y) + (3 + z )

x + y + z = 8 – 3 = 5 (x , y, z being naturals numbers)

Take 5 ones 1 _ 1 _ 1 _ 1 _ 1 , + symbol should be added in any 2 slots. (There are 4 empty slots)

This is done in ^{4}C_{2} ways = 6 ways.

Alternative method is to list down the possibilities as done below.

The question is **"In how many ways can 8 identical pens be distributed among Amal, Bimal, and Kamal so that Amal gets at least 1 pen, Bimal gets at least 2 pens, and Kamal gets at least 3 pens? [TITA]" **

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