Question

Find the third term of${\left(x+y\right)}^6$

Short answer

The third of the expansion is$15x^4{y}^2$

Step-by-step solution

Step 1. Use the r+1st term formula.

The ${\left(r+1\right)}^{st}$term in the binomial expansion of ${\left(a+b\right)}^n$is $\left(\begin{array}{c}n\\ r\end{array}\right)a^{n-r}{b}^r$

Substitute in the values into the binomial expansion,

$$\begin{gathered} a=x \\ b=y \\ n=6 \end{gathered}$$

Step 2. Substitute the values into the formula and simplify.

We are looking for the third term.

since $r+1=3$

then $r=2$.

Simply the expression,

$$\begin{gathered} {\left(x+y\right)}^6=\left(\begin{array}{c}6\\ 2\end{array}\right)x^{6-2}{y}^2 \\ =\frac{6!}{2!4!}{x}^4{y}^2 \\ =15x^4{y}^2 \end{gathered}$$

Step 3. Conclusion.

Therefore, the third term of the given expression is$15x^4{y}^2$