- In how many ways can you place 20 identical balls into five different boxes?
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:**Problem 7: Distribution of Identical Balls into Different Boxes**
*Question:* In how many ways can you place 20 identical balls into five different boxes?
In this problem, the task is to determine the number of ways to distribute 20 identical balls across five distinct boxes. This is a classic example of a combinatorial problem, often tackled using the "stars and bars" method in combinatorics. This approach involves calculating the number of non-negative integer solutions to the equation \(x_1 + x_2 + x_3 + x_4 + x_5 = 20\), where each \(x_i\) represents the number of balls in the \(i^{th}\) box.
Expert Solution
Step 1
Case 1:
No box remains empty.
The number of ways of distributing n identical objects among r persons such that each person gets at least one object is same as the number of ways of selecting (r−1) places out of (n−1) different places, that is n-1Cr-1.
Here, n= 20 and r=4
= 20-1C4-1 = 19C3 = 969.
Hence, there are 969 ways of placing 20 identical balls into five different boxes such that no box is empty.
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