#2**+15 **

I don't remember ever doing one like this before

I will give it a shot :)

$$\\(a+2b+3c)^6\\\\

=(a+[2b+3c])^6\\\\

=\displaystyle\sum_{r=0}^{6}^6C_r(a^r)(2b+3c)^{6-r}\\\\

$Now the degree of a must be 1 so I am only interested in terms where $ r=1\\\\

^6C_1(a^1)(2b+3c)^{5}\\\\

(6a)*[(2b+3c)^{5}]\\\\$$

$$\\$The general term for this is $\\\\

6a*[ ^5C_k*(2b)^k(3c)^{5-k}]\\\\

$The power of b must be 2 so k=2\\\\

6a*[ ^5C_2*(2b)^2(3c)^{3}]\\\\

6a*[ 10*4b^2*27c^3]\\\\

6a*[ 10*4b^2*27c^3]\\\\

6x*[1080*b^2c^3]\\\\

6480xb^2c^3\\\\

$So the coefficient of $ bc^2c^3 $ is 6480$$

I assume Chris got his answer from Wolfram|Alpha ?

Anyway they are the same so we are on a roll.

Melody
May 7, 2015

#2**+15 **

Best Answer

I don't remember ever doing one like this before

I will give it a shot :)

$$\\(a+2b+3c)^6\\\\

=(a+[2b+3c])^6\\\\

=\displaystyle\sum_{r=0}^{6}^6C_r(a^r)(2b+3c)^{6-r}\\\\

$Now the degree of a must be 1 so I am only interested in terms where $ r=1\\\\

^6C_1(a^1)(2b+3c)^{5}\\\\

(6a)*[(2b+3c)^{5}]\\\\$$

$$\\$The general term for this is $\\\\

6a*[ ^5C_k*(2b)^k(3c)^{5-k}]\\\\

$The power of b must be 2 so k=2\\\\

6a*[ ^5C_2*(2b)^2(3c)^{3}]\\\\

6a*[ 10*4b^2*27c^3]\\\\

6a*[ 10*4b^2*27c^3]\\\\

6x*[1080*b^2c^3]\\\\

6480xb^2c^3\\\\

$So the coefficient of $ bc^2c^3 $ is 6480$$

I assume Chris got his answer from Wolfram|Alpha ?

Anyway they are the same so we are on a roll.

Melody
May 7, 2015

#3**+10 **

Yep....I cheated....your method was far more impressive.....!!!!

This is a GOOD ONE to learn from....!!!!

CPhill
May 7, 2015

#4**+10 **

Thanks Chris,

I have taken your lead and put it in the 'great answers to learn" from thread

Melody
May 8, 2015

#5**0 **

Here's an addendum to this :

Let a = m and [ 2b + 3c] = n

So we are looking for the term in (m + n)^6 where a = m has a power of 1....and this is given by

C(6,1) mn^5 = 6an^5

So....we have

6a [ 2b + 3c]^5

And now we are looking for the term in (2b + 3c)^5 where [2b] has a power of 2....and this is given by:

C(5, 2) [2b]^2 [3b]^3 = 10 * 4b^2 * 27c^3 = 1080 b^2 c^3

Putting this all together, the term we are looking for is given by

6a [ 1080 b^2 c^3] = 6480ab^2c^3 ....... and 6480 is the coefficient

CPhill
Dec 26, 2016