Question:
The number of ways of distributing 5 identical things to 4 persons so that any person may get at most 5 things is:
The number of ways of distributing 5 identical things to 4 persons so that any person may get at most 5 things is:
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When the upper limit constraint on a variable is greater than or equal to the total sum required, you can ignore the upper limit and treat it as a standard "non-negative integer solutions" problem.
Updated On: Jul 18, 2026
- \( 84 \)
- \( 56 \)
- \( 14 \)
- \( 126 \)
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The Correct Option is B
Solution and Explanation
Concept:
The distribution of \( n \) identical items among \( r \) distinct persons is given by the formula for combinations with repetition (stars and bars).
• Formula: \( ^{n+r-1}C_{r-1} \)
• Constraints: Here \( x_1 + x_2 + x_3 + x_4 = 5 \) with \( x_i \geq 0 \).
Step 1: Analyzing the constraint.
The problem states each person gets "at most 5 things." Since the total number of things is exactly 5, it is impossible for any one person to get more than 5 things. Thus, the constraint \( x_i \leq 5 \) is naturally satisfied for all non-negative integer solutions.
Step 2: Applying the formula.
Here \( n = 5 \) (things) and \( r = 4 \) (persons). \[ \text{Number of ways} = ^{5+4-1}C_{4-1} = ^8C_3 \]
Step 3: Calculating the value.
\[ ^8C_3 = \frac{8 \times 7 \times 6}{3 \times 2 \times 1} = 8 \times 7 = 56 \]
• Formula: \( ^{n+r-1}C_{r-1} \)
• Constraints: Here \( x_1 + x_2 + x_3 + x_4 = 5 \) with \( x_i \geq 0 \).
Step 1: Analyzing the constraint.
The problem states each person gets "at most 5 things." Since the total number of things is exactly 5, it is impossible for any one person to get more than 5 things. Thus, the constraint \( x_i \leq 5 \) is naturally satisfied for all non-negative integer solutions.
Step 2: Applying the formula.
Here \( n = 5 \) (things) and \( r = 4 \) (persons). \[ \text{Number of ways} = ^{5+4-1}C_{4-1} = ^8C_3 \]
Step 3: Calculating the value.
\[ ^8C_3 = \frac{8 \times 7 \times 6}{3 \times 2 \times 1} = 8 \times 7 = 56 \]
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