Q.If $(1 + x)^{2n+1} = \sum_{r=0}^{2n+1} \binom{2n+1}{r} x^r$ , then $\sum_{r=0}^{n} (-1)^r \binom{2n+1}{r}$ equals

(-1)^n \binom{2n}{n}

(-1)^n \binom{2n+1}{n}

(-1)^{n+1} \binom{2n}{n}

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The correct solution involves applying the fundamental concept to derive the final value step by step...

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Q.If (1+x)2n+1=∑r=02n+1(2n+1r)xr(1 + x)^{2n+1} = \sum_{r=0}^{2n+1} \binom{2n+1}{r} x^r(1+x)2n+1=∑r=02n+1​(r2n+1​)xr, then ∑r=0n(−1)r(2n+1r)\sum_{r=0}^{n} (-1)^r \binom{2n+1}{r}∑r=0n​(−1)r(r2n+1​) equals