Answer:
The third term is [tex]24x^2y^2[/tex]
Step-by-step explanation:
The formula used to find the third term of the expansion (2x+y)^4 is called Binomial Theorem
The Binomial Theorem is:
[tex](x+a)^n = \sum_{k=0}^{n} {n \choose k}x^ka^{n-k}\\[/tex]
In the given question x = 2x
a = y
n = 4
We have to find the third term, so value of k will be 2 as k starts from 0
Putting the values in the Binomial Theorem
[tex]= {4 \choose 2}(2x)^2(y)^{4-2}\\= {4 \choose 2}4x^2(y)^{2}[/tex]
[tex]{n \choose k}==\frac{n!}{k!(n-k)!}[/tex]
Putting the values:
[tex]= {4 \choose 2}4x^2(y)^{2}\\=\frac{4!}{2!(4-2)!}4x^2(y)^{2}\\=\frac{4!}{2!2!}4x^2y^{2}\\=\frac{4*3*2*1}{2*2}4x^2y^{2}\\=\frac{24}{4}4x^2y^{2}\\=6*4x^2y^{2}\\=24x^2y^2[/tex]
So, the third term is [tex]24x^2y^2[/tex]
