The coefficient of $x^{6}$ in { $(1+x)^{6}$ + $(1+x)^{7}$ + ... + $(1+x)^{15}$ } is?

Question

The coefficient of $x^{6}$ in { $(1+x)^{6}$ + $(1+x)^{7}$ + ... + $(1+x)^{15}$ } is ?

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Options:

a) 16C9

b) 16C5 - 6C5

c) (16C6) - 1

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Now I have the answer as:

6C6 + 7C6 + ..... + 15C6

which I got after taking the coefficients from each term.

I am unable to express this answer in the terms of the options given. How do I do that?

Answer 1

Obviously, we have $$(1+x)^{6} + (1+x)^{7} + \ldots + (1+x)^{15}=\frac{(1+x)^{16}-(1+x)^{6}}x,$$ so we need the coefficient of $x^7$ of the numerator. That can come only from the first term and is $\binom{16}7=\binom{16}9,$ so it's option a).

Answer 2

HINT

$$(1+x)^6+...+(1+x)^{15}=\frac{(1+x)^6(1-(1+x)^{10})}{1-1-x}$$ $$=\frac{(1+x)^{16}-(1+x)^6}{x}$$

The above is by using geometric series sum.