In how many ways can 8 identical apples be divided among 3 sisters?

Options

A. 25
B. 65
C. 45
D. 24

Answer

To solve the problem of how many ways 8 identical apples can be divided among 3 sisters, we can use the "stars and bars" theorem in combinatorics. Here’s a step-by-step solution:

### Step 1: Identify the Variables
We have:
- **n** = number of identical items (apples) = 8
- **r** = number of recipients (sisters) = 3

### Step 2: 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 the formula:

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

### Step 3: Substitute the Values
Substituting the values we have:
- n = 8
- r = 3

So, we calculate:

\[ \text{Number of ways} = \binom{8 + 3 - 1}{3 - 1} = \binom{10}{2} \]

### Step 4: Calculate the Binomial Coefficient
Now, we need to calculate \(\binom{10}{2}\):

\[ \binom{10}{2} = \frac{10!}{2!(10-2)!} = \frac{10 \times 9}{2 \times 1} = \frac{90}{2} = 45 \]

### Step 5: Conclusion
Thus, the number of ways to distribute 8 identical apples among 3 sisters is **45**.