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. |
|
|
b2 b2□□□□□ Shuffle to get
other distributions. |
|
|
b2 b1
b1□□□□ Shuffle to get
other distributions. |
|
|
b1 b1 b1 b1□□□ Shuffle to get
other distributions. |
|
=42
=21
∴ the total number of ways = 7 + 42 + 21 + 105 + 35 = 210.