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 b_{n} stand for a bag contains n marbles, and b_{3
}b_{1}□□□□□
stand for the distribution that the 1st child gets a bag of 3 marbles, the 2^{nd}
child gets a bag of 1 marble, the other 5 children do
not get any.
Bags distribution |
#ways |
b_{4}□□□□□□ Shuffle to get
other distributions. |
_{} |
b_{3 }b_{1}□□□□□ Shuffle to get
other distributions. |
_{}=42 |
b_{2} b_{2}□□□□□ Shuffle to get
other distributions. |
_{}=21 |
b_{2} b_{1}
b_{1}□□□□ Shuffle to get
other distributions. |
_{} |
b_{1} b_{1} b_{1} b_{1}□□□ Shuffle to get
other distributions. |
_{} |
∴ the total number of ways = 7 + 42 + 21 + 105 + 35 = 210.