Problem 6

 

Find the number of ways to give 4 identical marbles to 7 children.

 

[Problem submitted by Steve Lee, LACC Professor of Mathematics.]

 

 

Solution for Problem 6:

 

We can put the marbles in bags and give the bags to the children.

Let bn stand for a bag contains n marbles, and b3 b1□□□□□ stand for the distribution that the 1st child gets a bag of 3 marbles, the 2nd child gets a bag of 1 marble, the other 5 children do not get any.

 

Bags distribution

#ways

b4□□□□□□

Shuffle to get other distributions.

b3 b1□□□□□

Shuffle to get other distributions.

=42

b2 b2□□□□□

Shuffle to get other distributions.

=21

b2 b1 b1□□□□

Shuffle to get other distributions.

b1 b1 b1 b1□□□

Shuffle to get other distributions.

the total number of ways = 7 + 42 + 21 + 105 + 35 = 210.