If the coefficient of middle term of the binomial expansion of `(1+x)^(2n)` is `alpha` and those of two middle terms of the binomial expansion of `(1+x)^(2n-1)` if `beta` and `gamma`, then which one of the following is correct?
(i) `alpha gt beta +gamma`
(ii) `alpha lt beta + gamma`
(iii) `alpha=beta+gamma`
(iv) `alpha = beta gamma`

Options

A. `alpha gt beta +gamma`
B. `alpha lt beta + gamma`
C. `alpha=beta+gamma`
D. `alpha = beta gamma`

Answer

To solve the problem step by step, we will analyze the coefficients of the middle terms in the binomial expansions of \((1+x)^{2n}\) and \((1+x)^{2n-1}\).

### Step 1: Coefficient of the middle term in \((1+x)^{2n}\)

The binomial expansion of \((1+x)^{2n}\) has \(2n + 1\) terms. The middle term is the \((n + 1)\)-th term. The coefficient of the middle term can be calculated using the binomial coefficient:

\[ \text{Coefficient of middle term} = \binom{2n}{n} \]

Let this coefficient be \(\alpha\):

\[ \alpha = \binom{2n}{n} \]