IF ` ""^(13)C_x = ""^(13)C_y ,` what is the value of x+y?

Options

A. 6
B. 7
C. 13
D. 26

Answer

To solve the problem where \( \binom{13}{x} = \binom{13}{y} \), we can use a property of combinations. Here are the steps to find the value of \( x + y \):

### Step 1: Understand the property of combinations
The property states that if \( \binom{n}{a} = \binom{n}{b} \), then either \( a = b \) or \( a + b = n \).

### Step 2: Apply the property to the given equation
In our case, we have \( n = 13 \), \( a = x \), and \( b = y \). Therefore, we can conclude:
1. \( x = y \) or
2. \( x + y = 13 \)

### Step 3: Analyze the implications
If \( x = y \), then \( x + y = 2x \). However, we are more interested in the case where \( x + y = 13 \) because it gives us a direct answer.

### Step 4: Conclusion
From the property, since \( x + y = 13 \) is one of the conditions when \( \binom{13}{x} = \binom{13}{y} \), we can conclude that:
\[ x + y = 13 \]

### Final Answer
Thus, the value of \( x + y \) is \( 13 \).

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