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# What is the coefficient of in ?

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978
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What is the coefficient of  in ?

Mellie  May 7, 2015

#2
+91469
+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
Sort:

#1
+81031
+10

6480......

Don't ask me how I know.....it just is.....!!!!!

CPhill  May 7, 2015
#2
+91469
+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
#3
+81031
+10

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

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

CPhill  May 7, 2015
#4
+91469
+10

Thanks Chris,

Melody  May 8, 2015
#5
+81031
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

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