Is there an equation that represents the nth row in Pascal's triangle?

Question

I'm doing binomial expansion and I'm rather confused at how people can find a certain coefficient of certain rows.

For example, if a problem was $(2x - 10y)^{54}$, and I were to figure out the $32^{\text{nd}}$ element in that expansion, how would I figure out?

Would I have to look at or draw out a Pascal's triangle, then go 1 by 1 until I hit row 54? Is there an equation that would tell me what the xth element of the nth row is by plugging in numbers?

Answer 1

The $n^{th}$ row reads

$$1,n,\frac{n(n-1)}2,\frac{n(n-1)(n-2)}{2\cdot3},\frac{n(n-1)(n-2)(n-3)}{2\cdot3\cdot4}\cdots$$

This is computed by recurrence very efficiently, like

$$1,54,\frac{54\cdot53}2=1431,\frac{1431\cdot52}3=24804,\frac{24804\cdot51}4=316251\cdots$$

Using symmetry, only the first half needs to be evaluated. Compared to the factorial formula, this is less prone to overflows.

Answer 2

Hint: $(a+b)^n=\sum\limits_{k=0}^n {n\choose k }a^kb^{n-k}$ where ${n\choose k}=\frac{n!}{k!(n-k)!}$