Coefficients Party, Part 1

Algebra Level 3

Find the coefficient of the $x^2y^6$ term in the expansion of $(3x-2y)^8.$

3 solutions

Jubayer Nirjhor
May 4, 2014

By Binomial Theorem, the answer is: $\dbinom{8}{2}\cdot 3^2\cdot (-2)^6=\fbox{16128}$

ur answer is right but the 8C6 comes instead of 8C2.

Harsh Bhavsar - 7 years, 1 month ago

$\dbinom{8}{6}=\dbinom{8}{2}$

$\dbinom{8}{6}=\frac{8!}{6!\times(8-6)!}=\frac{8!}{6!2!}$

$\dbinom{8}{2}=\frac{8!}{2!(8-2)!}=\frac{8!}{2!6!}=\frac{8!}{6!2!}$

Therefore,

$\dbinom{8}{6}=\dbinom{8}{2}$

In general,

$\dbinom{n}{r}=\dbinom{n}{n-r}$

Alex Segesta - 7 years, 1 month ago

both are same

Prajwal Kavad - 7 years ago

Binomial Theorem? Can you tell me more about that, please? I don't know anything about that.

Selene (Elly) Kirkland - 7 years, 1 month ago

See here: http://www.artofproblemsolving.com/Wiki/index.php/Binomial_Theorem

Hahn Lheem - 7 years, 1 month ago

it is a theorem used for finding expansions of expressions having very high exponents.

ishan pradhan - 6 years, 7 months ago

Leonhard Euler
May 9, 2014

Taking for granted that $(a + b)^n = \sum_{k=1}^n {n \choose k}a^{n-k} b^k$

From the problem we see that $n = 8, a = 3x, b = -2y$

Substituting and carrying out the computation will give the correct answer. A great first look on the binomial theorem is the following book

https://archive.org/details/philtrans01793630

Good explain... you should continue...

Heder Oliveira Dias - 7 years, 1 month ago

Pragya Singhal
May 6, 2014

using binomial expansion to solve the problem, STEP 1 :- (3x-2y)^8 STEP 2 :- {c(8,2)} x {(3)^2} x {(-2)^6} STEP 3 :- 16128