Question

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

Answer 1

Use the Binomial Theorem: `210x^4y^6`

Explanation:

Using the Binomial Theorem, you can quickly calculate the 4th term (or any `k^("th")` term).

The Binomial Theorem states that any binomial of the form `(x+a)^v` can be expanded to
`sum_(k=0)^infty ((v),(k))x^ka^(v-k)`

In trying to find the 4th term, we let `k=4` and in the binomial `(x+y)^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`