If n is even and
`""^(n)C_(0)lt""^(n)C_(1) lt ""^(n)C_(2) lt ....lt ""^(n)C_(r) gt ""^(n)C_(r+1) gt""^(n)C_(r+2) gt......gt""^(n)C_(n)`, then, r=

Options

A. `(n)/(2)`
B. `(n-1)/(2)`
C. `(n-2)/(2)`
D. `(n+2)/(2)`

Answer

To solve the problem, we need to analyze the given inequalities involving binomial coefficients. The question states that if \( n \) is even, then:

\[ \binom{n}{0} < \binom{n}{1} < \binom{n}{2} < \ldots < \binom{n}{r} > \binom{n}{r+1} > \binom{n}{r+2} > \ldots > \binom{n}{n} \]

We are required to find the value of \( r \).

### Step 1: Understand the behavior of binomial coefficients

The binomial coefficient \( \binom{n}{r} \) represents the number of ways to choose \( r \) elements from \( n \) elements. The values of \( \binom{n}{r} \) increase as \( r \) increases from \( 0 \) to \( \frac{n}{2} \) and then decrease as \( r \) increases from \( \frac{n}{2} \) to \( n \).

### Step 2: Identify the maximum value

Since \( n \) is even, the maximum value of \( \binom{n}{r} \) occurs at \( r = \frac{n}{2} \). This means:

\[ \binom{n}{0} < \binom{n}{1} < \ldots < \binom{n}{\frac{n}{2}} > \binom{n}{\frac{n}{2}+1} > \ldots > \binom{n}{n} \]

### Step 3: Relate to the problem statement

From the problem statement, we see that the coefficients are increasing up to \( r \) and then decreasing afterward. This indicates that \( r \) must be the point just before the maximum, which is \( \frac{n}{2} \).

### Step 4: Conclude the value of \( r \)

Thus, we conclude that:

\[ r = \frac{n}{2} \]

### Final Answer

The value of \( r \) is:

\[ \boxed{\frac{n}{2}} \]