Answer:
[tex]24x^{2}y^2[/tex]
Step-by-step explanation:
[tex]\boxed{\begin{minipage}{5cm} \underline{Binomial Theorem}\\\\$\displaystyle (a+b)^n=\sum^{n}_{k=0}\binom{n}{k} a^{n-k}b^{k}$\\\\\\where \displaystyle \binom{n}{k} = \frac{n!}{k!(n-k)!}\\\end{minipage}}[/tex]
We can use the Binomial Theorem to find any term of a binomial expansion.
The first term is when k = 0, so the third term is when k = 2.
Compare the given expression (x + 2y)⁴ with the formula to find the values of a, b and n.
Therefore:
Substitute the values into the formula to find the third term:
[tex]\implies \displaystyle\binom{4}{2}x^{4-2}(2y)^2[/tex]
[tex]\implies \dfrac{4!}{2!(4-2)!}x^{2}2^2y^2[/tex]
[tex]\implies \dfrac{4 \times 3\times \diagup\!\!\!\!2\times \diagup\!\!\!\!1}{2\times 1\times \diagup\!\!\!\!2\times \diagup\!\!\!\!1}\;x^{2}4y^2[/tex]
[tex]\implies \dfrac{12}{2}\:x^24y^2[/tex]
[tex]\implies 6x^{2}4y^2[/tex]
[tex]\implies 24x^{2}y^2[/tex]
