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

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.


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

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

Tags:

Triangles

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