Question:
In how many ways can 5 identical balls be distributed into 3 distinct boxes?
In how many ways can 5 identical balls be distributed into 3 distinct boxes?
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"Stars and bars" counts how many ways $n$ identical items can be split among $k$ groups using the formula $\binom{n+k-1}{k-1}$.
Updated On: Aug 1, 2025
- 15
- 21
- 25
35
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The Correct Option is B
Solution and Explanation
- Step 1: Recognizing the problem type - This is a "stars and bars" problem (distributing identical objects into distinct boxes).
- Step 2: Formula - Number of non-negative integer solutions to: \[ x_1 + x_2 + x_3 = 5 \] is: \[ \binom{n+k-1}{k-1} = \binom{5+3-1}{3-1} = \binom{7}{2} \]
- Step 3: Calculating - \[ \binom{7}{2} = \frac{7 \times 6}{2 \times 1} = 21 \]
- Step 4: Conclusion - There are 21 ways, matching option (2).
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