If `""^(50)C_(r)`= `""^(50)C_(r+2)`,find r.

Options

A. 24
B. 23
C. 22
D. 21

Answer

To solve the equation \( \binom{50}{r} = \binom{50}{r+2} \), we can use the property of combinations that states:

\[ \binom{n}{k} = \binom{n}{n-k} \]

This means that \( \binom{50}{r} = \binom{50}{50-r} \).

### Step 1: Set up the equation
Given:
\[ \binom{50}{r} = \binom{50}{r+2} \]

### Step 2: Use the property of combinations
Using the property mentioned above, we can rewrite the right-hand side:
\[ \binom{50}{r+2} = \binom{50}{50 - (r+2)} = \binom{50}{48 - r} \]

### Step 3: Set the two expressions equal
Now we have:
\[ \binom{50}{r} = \binom{50}{48 - r} \]

### Step 4: Apply the property again
Using the property again:
\[ r = 48 - r \]

### Step 5: Solve for \( r \)
Now, we can solve for \( r \):
\[ r + r = 48 \]
\[ 2r = 48 \]
\[ r = 24 \]

### Final Answer
Thus, the value of \( r \) is:
\[ \boxed{24} \]