Find the sum of the last eight coefficients in the expansion of ${(1 + x)}^{15}$

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Solution

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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