Question

The number of ways to distribute 10 identical apples among 4 children such that every child receives at least one apple is:

• A. \( {}^{13}C_3 \)
• B. \( {}^9C_3 \)
• C. \( {}^{10}C_4 \)
• D. \( {}^{14}C_4 \)

Hint:

To remember the "at least one" case, imagine placing 10 stars in a row. There are 9 gaps between them. To divide them into 4 groups, you need to choose 3 gaps to place "bars." Hence, \( {}^9C_3 \).

Answer 1

The Correct Option is B

Concept: This is a problem of distributing identical items into distinct bins, often solved using the "Stars and Bars" method.
• Formula for distributing \(n\) identical items among \(r\) people where each gets \(\geq 0\): \( {}^{n+r-1}C_{r-1} \)
• Formula where each gets \(\geq 1\): \( {}^{n-1}C_{r-1} \)

Step 1: Satisfying the initial constraint.
Since every child must receive at least one apple, we first give 1 apple to each of the 4 children. \[ \text{Apples remaining} = 10 - 4 = 6 \]

Step 2: Distributing the remaining apples.
Now we need to distribute the remaining 6 identical apples among the 4 children where any child can receive zero or more of these remaining apples. Here, \(n = 6\) and \(r = 4\). Using the formula \( {}^{n+r-1}C_{r-1} \): \[ {}^{6+4-1}C_{4-1} = {}^9C_3 \]

Step 3: Direct Application of the formula.
Alternatively, applying the "at least one" formula \( {}^{n-1}C_{r-1} \) directly where \(n=10\) and \(r=4\): \[ {}^{10-1}C_{4-1} = {}^9C_3 \]