The sum of the last eight coefficients in the expansion of ${(1 + x)}^{15}$ , is ?

2^{16}

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2^{15}

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2^{14}

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none of these

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Solution

The correct option is C $2^{14}$
Let Sum of last eight coefficients $= S$
$S = {15_{C}}_{8} + {15_{C}}_{9} + {15_{C}}_{10} + . . . + {15_{C}}_{15}$ .... $(1)$
Since ${n_{C}}_{r} = {n_{C}}_{n - r}$ we have ${15_{C}}_{15} = {15_{C}}_{0}$ , ${15_{C}}_{10} = {15_{C}}_{5}$ , ${15_{C}}_{9} = {15_{C}}_{6}, . .$ Using this
$S = {15_{C}}_{7} + {15_{C}}_{6} + {15_{C}}_{5} + . . . + {15_{C}}_{0}$ .... $(2)$
Adding eqns $(1)$ and $(2)$ we get
$2 S = {15_{C}}_{0} + {15_{C}}_{1} + {15_{C}}_{2} + {15_{C}}_{3} + . . . + {15_{C}}_{10} + {15_{C}}_{11} + {15_{C}}_{12} + {15_{C}}_{13} + {15_{C}}_{14} + {15_{C}}_{15}$
Using the concept that sum of the binomial coefficients in the expansion of ${(1 + x)}^{n} = 2^{n}$
$∴ 2 S = 2^{15}$ where $n = 15$
Hence $S = \frac{2^{15}}{2} = 2^{14}$

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