`"^(n)C_(r)=^(n)C_(n-r)`

Answer

To prove the identity \( \binom{n}{r} = \binom{n}{n-r} \), we will follow these steps:

### Step 1: Write the formula for \( \binom{n}{r} \)

The binomial coefficient \( \binom{n}{r} \) is defined as:

\[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \]

### Step 2: Write the formula for \( \binom{n}{n-r} \)

Now, let's express \( \binom{n}{n-r} \):

\[ \binom{n}{n-r} = \frac{n!}{(n-r)!(n-(n-r))!} \]

### Step 3: Simplify \( n - (n-r) \)

Now, simplify \( n - (n-r) \):

\[ n - (n-r) = n - n + r = r \]

### Step 4: Substitute back into the formula

Substituting \( n - (n-r) \) back into the formula for \( \binom{n}{n-r} \):

\[ \binom{n}{n-r} = \frac{n!}{(n-r)!r!} \]

### Step 5: Compare the two expressions

Now we have:

\[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \]
\[ \binom{n}{n-r} = \frac{n!}{(n-r)!r!} \]

### Step 6: Conclude the proof

Since both expressions are equal, we conclude that:

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

Thus, we have proved that \( \binom{n}{r} = \binom{n}{n-r} \).

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