Question

The coefficient of x^4 in the expansion of (2 + x)^13 is

Answer 1

Answer:

Step-by-step explanation:

The given binomial is $(2+x)^{13}$

$\sf{\bold{The\:\:general\:\:term\:\:on\:\:expanding\:\:it\:\:is,}}$

$\sf{t_{r+1}=\:^{13}C_{r}\cdot\:{2}^{13-r}\cdot\:{x}^{r}}$

$\tt{\blue{To\:\:obtain\:\:the\,\,coefficient\,\,of\,\,{x}^{4}\,\, ,put\,\,r=4\,\,in\,\,above\,\,equation}}$

So,

$\sf{t_{4+1}=\:^{13}C_{4}\cdot\:{2}^{13-4}\cdot\:{x}^{4}}$

So,

$\sf{t_{5}=\:^{13}C_{4}\cdot\:{2}^{9}\cdot\:{x}^{4}}$

$\sf{t_{5}=\dfrac{13\cdot12\cdot11\cdot10}{4\cdot3\cdot2\cdot\cdot1}\cdot\:512\cdot\:{x}^{4}}$

$\sf{t_{5}=\dfrac{13\cdot11\cdot10}{2\cdot1}\cdot\:512\cdot\:{x}^{4}}$

$\sf{t_{5}=13\cdot11\cdot5\cdot\:512\cdot\:{x}^{4}}$

$\sf{t_{5}=366080{x}^{4}}$

$\tt{\pink{Hence,\,\,coefficient\,\,of\,\,{x}^{4}\,\, is\,\,366080}}$