In attempting to show that for all

MATHEMATICS · Single correct · CBSE · Mathematics · Binomial Theorem · Binomial Theorem

Problem

In attempting to show that

\binom{2n}{n} \le 4^{\,n}

for all

n \ge 1

, which of the following identities is required to relate

\binom{2n+2}{\,n+1}

\binom{2n}{\,n}

in the inductive step? (a)

\displaystyle \binom{2n+2}{\,n+1} = \frac{(2n+2)}{(n+1)}\,\binom{2n}{\,n}

(b)

\displaystyle \binom{2n+2}{\,n+1} = \frac{(2n+2)(2n+1)}{(n+1)(n+1)}\,\binom{2n}{\,n}

(c)

\displaystyle \binom{2n+2}{\,n+1} = \frac{(2n+1)}{(n+1)}\,\binom{2n}{\,n}

(d)

\displaystyle \binom{2n+2}{\,n+1} = \frac{(2n+2)(2n+1)}{(n+1)}\,\binom{2n}{\,n}