Counting sets a Find the coefficient of anb2n in the expansi

(Counting sets)

a) Find the coefficient of a^nb^2n in the expansion of (a+2b)^3n

b) Find the coefficient of x^5 in the expansion of (x+3)^5n

c) Find the the coefficient of x^5 in the expansion of (2+x^2)^5n

Solution

a) A general terms in the expansion is

a^k b^(3n-k)2^{3n-k}C(3n,k)

So here we need k=n

So,

Coefficient is

C(3n,n)2^{2n}

b)

General term is :x^k 3^{5n-k} C(5n,k)

So coefficient of x^5 is

3^{5n-5}C(5n,5)

c)

General term is

(x^2)^k2^(5n-k)C(5n,k)

x^(2k)2^(5n-k)C(5n,k)

We see that each term has even power of x

So coefficient of x^5 is 0