The total number of ways in which 11 identical apples can be distributed among 6 children such that every student gets atleast one apple,is

Options

A. 252
B. 462
C. 42
D. none of these

Answer

To solve the problem of distributing 11 identical apples among 6 children such that each child gets at least one apple, we can follow these steps:

### Step 1: Ensure Each Child Gets One Apple
Since each of the 6 children must receive at least one apple, we start by giving one apple to each child. This means we will distribute 6 apples initially.

**Calculation:**
- Apples distributed: 6 (1 apple to each of the 6 children)
- Remaining apples: 11 - 6 = 5 apples

### Step 2: Distribute Remaining Apples
Now, we need to distribute the remaining 5 apples among the 6 children. Since the apples are identical, we can use the "stars and bars" theorem to find the number of ways to do this.

### Step 3: Apply Stars and Bars Theorem
The stars and bars theorem states that the number of ways to distribute \( n \) identical items (stars) into \( r \) distinct groups (bars) is given by the formula:

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

In our case:
- \( n = 5 \) (remaining apples)
- \( r = 6 \) (children)

### Step 4: Calculate the Combinations
Now we can substitute the values into the formula:

\[ \text{Number of ways} = \binom{5 + 6 - 1}{6 - 1} = \binom{10}{5} \]

### Step 5: Compute the Binomial Coefficient
Now we calculate \( \binom{10}{5} \):

\[ \binom{10}{5} = \frac{10!}{5! \cdot (10 - 5)!} = \frac{10!}{5! \cdot 5!} = \frac{10 \times 9 \times 8 \times 7 \times 6}{5 \times 4 \times 3 \times 2 \times 1} = 252 \]

### Final Answer
Thus, the total number of ways to distribute 11 identical apples among 6 children such that each child gets at least one apple is **252**.

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