Question

In how many ways can 5 identical balls be distributed into 3 distinct boxes? 
 

• A. 15
• B. 21
• C. 25
• D. 35

Hint:

"Stars and bars" counts how many ways $n$ identical items can be split among $k$ groups using the formula $\binom{n+k-1}{k-1}$.

Answer 1

The Correct Option is B

- 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).