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

1 Answer
Feb 18, 2017

Answer:

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.

http://www.slideshare.net/vbhunt/pascals-triangle-31417851

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#