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

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Answer 1 · 2017-02-18T02:44:07

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..

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

![http://www.slideshare.net/vbhunt/pascals-triangle-31417851](useruploads.socratic.org)

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

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