How to find the coefficient of x^6y^8 in the Binomial Expansion of (2x^3-y^2)^6 ?

Q: How to find the coefficient of x^6y^8 in the Binomial Expansion of (2x^3-y^2)^6 ?

Answer :

Nth term in the Binomial Expansion of (2x^3-y^2)^6 will be given as
\[=>\ \frac{n!}{k!\left(n-k\right)!}\left(2x^3\right)^{n-k}\left(-y^2\right)^k\]\[=>\frac{n!}{\ k!\left(n-k\right)!}\left(-1\right)^k2^{n-k}x^{3n-3k}y^{2k}\]We want the coefficient of x^6*y^8 that is
\[=>3n-3k=6,\ 2k=8\]\[=>k=4,\ n=6\]Substituting n =6, k= 4 into the formula
\[\frac{n!}{\ k!\left(n-k\right)!}\left(-1\right)^k2^{n-k}x^{3n-3k}y^{2k}\]\[=>\frac{n!}{\ k!\left(n-k\right)!}\left(-1\right)^k2^{n-k}x^6y^8\]\[=>\frac{6!}{4!\left(6-4\right)!}\left(-1\right)^42^{6-4}x^6y^8\]\[=>\frac{4\ast6!}{4!\ast2!}x^6y^8\]\[=>60x^6y^8\]Therefore the coefficient of x^6y^8 in the Binomial Expansion of ((2*x^3-y^2)^6) is 60.