Question

How do you evaluate `((8), (7))` using Pascal's triangle?

Answer 1

`((8),(7))=8`

Explanation:

When I have a sum of the sort `((n),(k)),` the number `n` corresponds to the row number of Pascal's Triangle, and `k` corresponds to the column number of that row, where the first column has the value `k=0`.

`((n),(k))` is typically read "`n` choose `k`." There is a formula to find it:

`((n),(k))=(n!)/(k!(n-k)!)`, where `!` is the factorial function.

So inputting the values for `n` and `k` in this situation gives:

`(8!)/(7!(8-7)!)`

`(8*cancel(7!))/(cancel(7!)*1!)`

`=8`

It is important to note that `((n),(n-1))=n`.

We can also use Pascal's Triangle:

Remember, the first row and column are given by `n=0` and `k=0` respectively.

Going to the eigth row and the seventh column gives us the same number, `8`.