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?


The $n^{th}$ row reads


This is computed by recurrence very efficiently, like


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


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


The nth row of a pascals triangle is:

$$_nC_0, _nC_1, _nC_2, ...$$

recall that the combination formula of $_nC_r$ is

$$ \frac{n!}{(n-r)!r!}$$

So element number x of the nth row of a pascals triangle could be expressed as

$$ \frac{n!}{(n-(x-1))!(x-1)!}$$

  • $\begingroup$ Welcome to MSE. Your answer adds nothing new to the already existing answers. $\endgroup$ – José Carlos Santos Dec 30 '18 at 19:46

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