+0  
 
0
42
6
avatar+90 

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

MathCuber  Aug 9, 2018
 #1
avatar+19841 
+3

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

heureka  Aug 9, 2018
edited by heureka  Aug 9, 2018
 #2
avatar+92927 
+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

Melody  Aug 9, 2018
 #3
avatar+19841 
+1

Thank you

 

laugh

heureka  Aug 9, 2018
 #4
avatar+92927 
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.

Melody  Aug 9, 2018
 #5
avatar+19841 
+1

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+92927 
+1

Thanks Heureka. That is great !!

Melody  Aug 10, 2018

23 Online Users

avatar

New Privacy Policy

We use cookies to personalise content and advertisements and to analyse access to our website. Furthermore, our partners for online advertising receive information about your use of our website.
For more information: our cookie policy and privacy policy.