# Binomial Expansion Help Needed!

#### arshinmar

I came across a question, which was:

Find the coefficient of c^4d^11 in the expansion of (2c+5d)(c+d)^14.

I can do it easily with the (c+d) bracket, but the (2c+5d) complicates it for me.

Could someone help me out? Thanks a lot!

#### romsek

Math Team
is this supposed to be

$[(2c+5d)(c+c)]^{14}$

or

$(2c+5d)(c+d)^{14}$

?

The latter.

#### romsek

Math Team
$(2c+5d)(c+d)^{14}$
well first expand out the right term

$(2c+5d) \displaystyle \sum_{k=0}^{14} \binom{14}{k} c^k d^{14-k}$

now do the multiplication

$\displaystyle 2\sum_{k=0}^{14}\binom{14}{k} c^{k+1} d^{14-k} + 5\displaystyle \sum_{k=0}^{14}\binom{14}{k} c^k d^{14-k+1}$

grab the $c^4 d^{11}$ coefficents from each sum

$\displaystyle 2 \binom{14}{3} + 5 \binom{14}{4}$

and I leave you to evaluate that last expression

2 people