`""^(17) C_r = ""^(17) C_(r+3)` find the value of r

Options

A. 17
B. 6
C. 7
D. 13

Answer

To solve the equation \( \binom{17}{r} = \binom{17}{r+3} \), we can use the property of combinations that states \( \binom{n}{k} = \binom{n}{n-k} \). This implies that if \( \binom{n}{k_1} = \binom{n}{k_2} \), then \( k_1 + k_2 = n \).

### Step-by-Step Solution:

1. **Identify the values**: In our case, we have \( n = 17 \), \( k_1 = r \), and \( k_2 = r + 3 \).

2. **Set up the equation**: According to the property mentioned, we can write:
\[ r + (r + 3) = 17 \]

3. **Simplify the equation**: Combine like terms:
\[ 2r + 3 = 17 \]

4. **Isolate the variable**: Subtract 3 from both sides:
\[ 2r = 17 - 3 \]
\[ 2r = 14 \]

5. **Solve for r**: Divide both sides by 2:
\[ r = \frac{14}{2} = 7 \]

### Final Answer:
Thus, the value of \( r \) is \( 7 \).

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