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Chapter 6: Problem 48

In how many ways can we place 10 identical balls in 12 boxes if each box can hold 10 balls?

Short Answer

Expert verified
There are 352716 ways to place 10 identical balls in 12 boxes, with each box able to hold up to 10 balls.

Step by step solution

01

Determine the problem type

Identify the problem as a combination problem with repetition. This is determined by the fact that there are more balls than the boxes and each box can contain more than one ball.
02

Use the stars and bars method

The stars and bars method is a common technique for solving this type of problem. Here, the balls are the stars and the boxes are the bars. The number of ways to distribute the balls is equivalent to the number of ways to arrange these stars and bars.
03

Calculate the number of arrangements

Use the combinatorial formula to calculate the total number of arrangements which is \( C(n+r-1, r-1) \) where \( n \) is the number of balls and \( r \) is the number of boxes. Here, \( n = 10 \) and \( r = 12 \). So, the number of arrangements is \( C(10+12-1, 12-1) = C(21, 11) \), which is the number of ways to choose 11 items from 21.
04

Computation

Compute \( C(21, 11) \) = \(\frac{21!}{11!(21-11)!}\) = 352716.

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