Question

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

Answer 1

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, `"_7C_4"` corresponds to the fifth value in the row, namely 35.

Note that `"_7C_4"` is equal to `"_7C_3"`, the fourth value in the row. Both are written

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