Question

The number of ways in which we can distribute n identical balls in k boxes is 

• A. \((n+k-1)C_{k-1}\)
• B. \((n)C_{k-1}\)
• C. \(^{(n-1)}C_{{k-1}}\)
• D. \((n+k-1)C_{k}\)
• E. \(nC_{k}\)

Answer 1

The Correct Option is A

Given: We need to distribute \( n \) identical balls into \( k \) distinct boxes.

The number of ways to distribute \( n \) identical items into \( k \) distinct boxes is given by the formula:

\[ \text{Number of ways} = \binom{n + k - 1}{k - 1} \]

This formula comes from the stars and bars theorem, where we represent the balls as stars and the dividers between boxes as bars. We need \( k - 1 \) bars to divide \( n \) stars into \( k \) groups.

Let's verify the options:

• (A) \( \binom{n}{k} \) - Incorrect, this would be for choosing \( k \) balls out of \( n \) distinct balls
• (B) \( \binom{n}{k-1} \) - Incorrect, similar issue as (A)
• (C) \( \binom{n + k - 1}{k - 1} \) - Correct, matches our formula
• (D) \( \binom{n - 1}{k - 1} \) - Incorrect, this would be for positive integer solutions
• (E) \( \binom{n + k}{k} \) - Incorrect, this would be if we had \( k \) additional items

 

Therefore, the correct answer is (C) \( \binom{n + k - 1}{k - 1} \).

Answer 2

The number of ways to distribute \( n \) identical balls into \( k \) distinct boxes is given by the stars and bars method. We have \( n \) identical balls (stars) and we want to divide them into \( k \) boxes. To do this, we need \( k-1 \) dividers (bars).

The number of ways to arrange \( n \) stars and \( k-1 \) bars is given by:

\[ \binom{n+k-1}{k-1} = \binom{n+k-1}{n} \]

Therefore, there are \( \binom{n+k-1}{k-1} \) ways to distribute \( n \) identical balls into \( k \) distinct boxes.