# How do you evaluate 7C4 using Pascal's triangle?

Feb 18, 2017

There will be a row in Pascal's triangle that will give a value to every combination "_7C_r, for $r = 0 \to 7$. Details follow..

#### Explanation:

Here is an image of Pascal's triangle having rows zero through 16.

Look at row seven. It contains the numbers

1...7...21...35...35...21...7...1

(I used dots as this was the only way I could find to space the terms appropriately.)

The values in this row are equal to "_7C_r where $r$ is the position of the number in the row, starting with $r = 0$.

So, $\text{_7C_4}$ corresponds to the fifth value in the row, namely 35.

Note that $\text{_7C_4}$ is equal to $\text{_7C_3}$, the fourth value in the row. Both are written

(7!)/((3!)(4!)) = 35