# How do you find the 4th term in the expansion of the binomial (x+y)^10?

May 14, 2017

Use the Binomial Theorem: $210 {x}^{4} {y}^{6}$

#### Explanation:

Using the Binomial Theorem, you can quickly calculate the 4th term (or any ${k}^{\text{th}}$ term).

The Binomial Theorem states that any binomial of the form ${\left(x + a\right)}^{v}$ can be expanded to
${\sum}_{k = 0}^{\infty} \left(\begin{matrix}v \\ k\end{matrix}\right) {x}^{k} {a}^{v - k}$

In trying to find the 4th term, we let $k = 4$ and in the binomial ${\left(x + y\right)}^{10}$, the term $a = y$ and $v = 10$. This gives us

((v),(k))x^ka^(v-k)=((10),(4))x^4y^(10-4)=(10!)/((10-4)!4!)x^4y^6
=(10!)/(6!4!)x^4y^6=210x^4y^6