Try something new - Expansion challenge.

Find the coefficient of x^2y^3z in the expansion of (4x + y - 2z)^6
The coefficient is calculated as follows:
[6!(original exponent) / 2!.3!.1!(exponents of the desired term)] X [4x(First term of original trinomial)]^(desired power(2)) X [y(Second term of original trinomial)]^(desired power(3)) X [2z(Third term of original trinomial)]^(desired power(1))
6! /2!.3!.1!  * (4x)^2 * (y^3) * (2z)=60 * 16x^2 * y^3 *2z=1920x^2y^3z

[Note: The number of terms of this expansion is calculated as follows:( 6(The exponent of the trinomial)) + number of terms - 1 nCr the exponent of the trinomial =6 + 3 -1 nCr 6 =8 nCr 6 =28 terms]

Thanks to my CAS system, here are ALL the 28 terms and their coefficients as calculated above:
4096 x^6 + 6144 x^5 y - 12288 x^5 z + 3840 x^4 y^2 - 15360 x^4 y z + 15360 x^4 z^2 + 1280 x^3 y^3 - 7680 x^3 y^2 z + 15360 x^3 y z^2 - 10240 x^3 z^3 + 240 x^2 y^4 - 1920 x^2 y^3 z + 5760 x^2 y^2 z^2 - 7680 x^2 y z^3 + 3840 x^2 z^4 + 24 x y^5 - 240 x y^4 z + 960 x y^3 z^2 - 1920 x y^2 z^3 + 1920 x y z^4 - 768 x z^5 + y^6 - 12 y^5 z + 60 y^4 z^2 - 160 y^3 z^3 + 240 y^2 z^4 - 192 y z^5 + 64 z^6
(28 terms)

Thanks guest,

It is great to see that my post sparked some interest. :)

The first one is great.

You did make one small error though. (It is a very common error all through mathematics) 

Can you find it?  

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The second one is also good. I haver not checked it I am just impressed that you did it.  I assume it is correct since you did it with a program.

However I do not understand your formula.  How can 6+3- a positive number =28

mmmm

Oh maybe my detective work has worked.... do you mean  ??? 

  nCr

where  n=6+3-1          [Exponent of polynomial + number of terms - 1]

and r = 6                     [exponent of polynomial]

So that is       8C6 = 28       Bingo   

So for a nomal cubic binomial  (a+b)^3    There would be (3+2-1)C3 = 4C3 = 4 terms. 

Das ist Ausgezeichnet!   

Hi Melody: Thanks for the compliment. Is it the (-) sign before the coefficient? 

Number of terms = The exponent of the trinomial =6 + Number of terms= 3 - 1=8C(the exponent of the trinomial) =6. Or, as you calculated: 8C6 =28.

Thanks again. I did think of both mistakes but was too lazy, especially the calculation of the number of terms of the expansion. I thought that was "known" to most people interested in Math! Now, I know better!. I didn't know you spoke German!!!.