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Find the coefficient of x^3 y^3 z^2 in the expansion of (x+y+z)^8.

 Aug 9, 2018
 #1
avatar+26364 
+5

Find the coefficient of x^3 y^3 z^2 in the expansion of (x+y+z)^8.

 

Trinomial Coefficient:

\(\begin{array}{|rcll|} \hline \dbinom{8}{3,3,2} &=& \dfrac{8!}{3!3!2!} \\\\ &=& \dfrac{3!4\cdot 5 \cdot 6 \cdot 7 \cdot 8}{3!3!2!} \\\\ &=& \dfrac{4\cdot 5 \cdot 6 \cdot 7 \cdot 8}{3!2!} \quad & | \quad 3! = 6 \quad 2! = 2 \\\\ &=& \dfrac{4\cdot 5 \cdot 6 \cdot 7 \cdot 8}{6\cdot 2} \\\\ &=& \dfrac{4\cdot 5 \cdot 7 \cdot 8}{2} \\\\ &=& 2\cdot 5 \cdot 7 \cdot 8 \\\\ &=& 10 \cdot 56 \\\\ &\mathbf{=}& \mathbf{ 560 } \\ \hline \end{array}\)

 

The trinomial coefficient of \(x^3 y^3 z^2\) in the expansion of \((x+y+z)^8\) is 560

 

laugh

 Aug 9, 2018
edited by heureka  Aug 9, 2018
 #2
avatar+118587 
+1

Thanks Heureka, I can't remember ever seeing this type of solution before. 

So I am very pleased that you have shown me.   laugh

 Aug 9, 2018
 #3
avatar+26364 
+2

Thank you

 

laugh

heureka  Aug 9, 2018
 #4
avatar+118587 
0

Hi Heureka,

I am wondering if it can still be done easily if it is made more complicated ?

 

eg

Find the coefficient of x^3 y^3 z^2 in the expansion of (3x+5y+7z)^8.

 Aug 9, 2018
 #5
avatar+26364 
+3

Hi Melody

"Find the coefficient of x^3 y^3 z^2 in the expansion of (3x+5y+7z)^8".

 

 

Set \(a = 3x\)

Set \(b = 5y\)

Set \(c = 7z\)

Set \(n = 8\)

Set \(i = 3\)

Set \(j = 3\)

Set \(k = 2\)

 

Trinomial Coefficient of  \(x^3 y^3 z^2\):
\(\begin{array}{|rcll|} \hline \dbinom{8}{3,3,2}(3x)^3\cdot(5y)^3\cdot (7z)^2 &=& \dfrac{8!}{3!3!2!} (3x)^3\cdot(5y)^3\cdot (7z)^2 \\\\ &=& \dfrac{8!}{3!3!2!} \cdot 3^35^37^2 x^3y^3z^2 \quad & | \quad \dfrac{8!}{3!3!2!} = 560 \\\\ &=& 560\cdot 3^35^37^2 x^3y^3z^2 \\\\ &=& 560\cdot 27 \cdot 125 \cdot 49 x^3y^3z^2 \\\\ &\mathbf{=}& \mathbf{ 92610000 x^3y^3z^2 } \\ \hline \end{array}\)

 

\(\text{The coefficient of $x^3 y^3 z^2$ in the expansion of $(3x+5y+7z)^8$ is $\mathbf{92~ 610~ 000}$ } \)

 

Source: https://en.wikipedia.org/wiki/Trinomial_expansion

 

laugh

heureka  Aug 10, 2018
edited by heureka  Aug 10, 2018
 #6
avatar+118587 
+2

Thanks Heureka. That is great !!

Melody  Aug 10, 2018

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