The number of ways in which 21 identical apples to be distributed into 3 chilred in such a way that each children get at least 2 apples is

Options

A. 133
B. 134
C. 135
D. 136

Answer

To solve the problem of distributing 21 identical apples among 3 children such that each child receives at least 2 apples, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Problem**: We have 21 identical apples and we need to distribute them to 3 children (let's call them Child 1, Child 2, and Child 3) with the condition that each child must receive at least 2 apples.

2. **Initial Distribution**: Since each child must receive at least 2 apples, we can start by giving 2 apples to each child. This means we will distribute:
\[ 2 \times 3 = 6 \text{ apples} \]
After this initial distribution, the number of apples left is:
\[ 21 - 6 = 15 \text{ apples} \]

3. **Reformulating the Problem**: Now, we need to distribute the remaining 15 apples among the 3 children without any restrictions (since they have already received their minimum of 2 apples). Let:
- \( x \) = number of additional apples given to Child 1
- \( y \) = number of additional apples given to Child 2
- \( z \) = number of additional apples given to Child 3

We need to solve the equation:
\[ x + y + z = 15 \]

4. **Using the Stars and Bars Theorem**: The problem of distributing \( n \) identical items (in this case, apples) into \( r \) distinct groups (in this case, children) can be solved using the "stars and bars" theorem. The number of ways to distribute \( n \) identical items into \( r \) groups is given by:
\[ \binom{n + r - 1}{r - 1} \]
In our case, \( n = 15 \) and \( r = 3 \). Therefore, we need to calculate:
\[ \binom{15 + 3 - 1}{3 - 1} = \binom{17}{2} \]

5. **Calculating the Combination**: Now we compute \( \binom{17}{2} \):
\[ \binom{17}{2} = \frac{17 \times 16}{2 \times 1} = \frac{272}{2} = 136 \]

6. **Conclusion**: Thus, the total number of ways to distribute the 21 identical apples among 3 children, ensuring each child gets at least 2 apples, is:
\[ \boxed{136} \]