Question

find the coefficient of x⁵ in the expansion of 1 + (1+x) + (1+x)²+ ... + (1+x)¹⁰​

Answer 1

Answer:

Step-by-step explanation:

We have,

$\tt{1+(1+x)+(1+x)^2+(1+x)^3+...+(1+x)^{10}}$

$\sf{x^5}$, will occur after the expansions of the terms of

$\sf{(1+x)^5,\,(1+x)^6,\,(1+x)^7,\,(1+x)^8,\,(1+x)^9,\,(1+x)^{10}}$

Now, the coefficient :

$\sf{\,^{5}C_{5}(x)^5+\,^{6}C_{5}(x)^5+\,^{7}C_{5}(x)^5+\,^{8}C_{5}(x)^5+\,^{9}C_{5}(x)^5+\,^{10}C_{5}(x)^5}$

$\sf{=\big(\,^{5}C_{5}+\,^{6}C_{5}+\,^{7}C_{5}+\,^{8}C_{5}+\,^{9}C_{5}+\,^{10}C_{5}\big)(x)^5}$

$\sf{=\bigg(1+6+7\times6+8\times7\times6+9\times8\times7\times6+10\times9\times8\times7\times6\bigg)(x)^5}$

$\sf{=\bigg(1+6+42+336+9\times336+10\times9\times336\bigg)(x)^5}$

$\sf{=\bigg(49+336+11\times9\times336\bigg)(x)^5}$

$\sf{=\bigg(385+11\times3024\bigg)(x)^5}$

$\sf{=\bigg(385+33264\bigg)(x)^5}$

$\sf{=\bigg(33649\bigg)(x)^5}$

Hence, the required coefficient = 33649