If `""^(15)C_(3r) `=` ""^(15)C_(r+3)`, find r.

Answer

To solve the equation \( \binom{15}{3r} = \binom{15}{r+3} \), we will use the property of combinations that states \( \binom{n}{k} = \binom{n}{n-k} \).

### Step-by-Step Solution:

1. **Set Up the Equation**:
We start with the equation:
\[ \binom{15}{3r} = \binom{15}{r+3} \]

2. **Apply the Combination Property**:
Using the property \( \binom{n}{k} = \binom{n}{n-k} \), we can rewrite \( \binom{15}{r+3} \):
\[ \binom{15}{r+3} = \binom{15}{15 - (r+3)} = \binom{15}{12 - r} \]
Thus, we have:
\[ \binom{15}{3r} = \binom{15}{12 - r} \]

3. **Equate the Indices**:
Since the combinations are equal, we can set the indices equal to each other:
\[ 3r = 12 - r \]

4. **Solve for \( r \)**:
Rearranging the equation gives:
\[ 3r + r = 12 \]
\[ 4r = 12 \]
Dividing both sides by 4:
\[ r = 3 \]

5. **Conclusion**:
The value of \( r \) is:
\[ \boxed{3} \]