Using Binomial Theorem, give the closed form expression for: \textstyle\sum_{k=0}^n∑ k=0 n ​ {n \choose k}( k n ​ )3n · 2k

Using Binomial Theorem, let us give the closed form expression for

$\sum\limits_{k=0}^n {n \choose k}3^n · 2^k.$

It follows that

$\sum\limits_{k=0}^n {n \choose k}3^n · 2^k =3^n\sum\limits_{k=0}^n {n \choose k}2^k1^{n-k}\\=3^n(2+1)^n=3^n3^n=3^{2n}=9^n.$

We conclude that the closed form expression is $9^n.$