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Question

If the sum of the series 12C5+7.12C6+21.12C7+35.12C8+35.12C9+21.12C10+7.12C11+12C12 is equal to 2n+1C12 then n is

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Solution

Series is 7C7.12C5+7C6.12C6+7C5.12C7+7C4.12C8+7C3.12C9+7C2.12C10+7C1.12C11+7C0.12C12
(1+x)7=7C0+7C1x+7C2x2++7C7x7
(1+x)12=12C0+12C1x+12C2x2+12C5x5+12C12x12
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required series is equal to coeff. of x12 in multiplication of above two series.

So coeff. of x12 in (1+x)19 is 19C12.

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