The number of ways in which thirty five apples can be distributed among 3 boys so that each can have any number of apples, is :

Options

A. 1332
B. 666
C. 333
D. none of these

Answer

To solve the problem of distributing 35 apples among 3 boys such that each boy can receive any number of apples (including zero), we can use the "stars and bars" theorem from combinatorics. Here’s a step-by-step solution:

### Step 1: Understand the Problem
We need to find the number of ways to distribute 35 identical apples (stars) among 3 distinct boys (groups). Each boy can receive zero or more apples.

### Step 2: Set Up the Equation
Let \( x_1, x_2, \) and \( x_3 \) represent the number of apples received by Boy 1, Boy 2, and Boy 3, respectively. We need to solve the equation:
\[ x_1 + x_2 + x_3 = 35 \]
where \( x_1, x_2, x_3 \geq 0 \).

### Step 3: Apply the Stars and Bars Theorem
According to the stars and bars theorem, the number of ways to distribute \( n \) identical items into \( r \) distinct groups is given by:
\[ \binom{n + r - 1}{r - 1} \]
In our case, \( n = 35 \) (the apples) and \( r = 3 \) (the boys).

### Step 4: Substitute into the Formula
Substituting the values into the formula, we have:
\[ \binom{35 + 3 - 1}{3 - 1} = \binom{37}{2} \]

### Step 5: Calculate the Binomial Coefficient
Now we calculate \( \binom{37}{2} \):
\[ \binom{37}{2} = \frac{37!}{2!(37 - 2)!} = \frac{37 \times 36}{2 \times 1} = \frac{1332}{2} = 666 \]

### Final Answer
Thus, the number of ways to distribute 35 apples among 3 boys is \( 666 \).