Question

The number of arrangements of six identical balls in three identical bins is

Hint:

The number of arrangements of identical items in identical bins can be determined by finding the number of distinct partitions of the total items.

Answer 1

Correct Answer: 7

This is a typical problem in combinatorics, where we need to find the number of distinct partitions of 6 identical balls into 3 identical bins. The problem of partitioning \(n\) identical objects into \(k\) identical boxes is equivalent to the number of solutions to the equation: \[ x_1 + x_2 + x_3 = 6 \] where \(x_1, x_2, x_3\) are non-negative integers representing the number of balls in each bin. The number of solutions to this equation is given by the number of distinct partitions of 6 into up to 3 parts. The valid partitions of 6 are: \[ (6,0,0), (5,1,0), (4,2,0), (4,1,1), (3,3,0), (3,2,1), (2,2,2) \] Thus, there are 7 distinct arrangements. The number of arrangements of six identical balls in three identical bins is: \[ \boxed{7} \]