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

Feb 12, 2018

$\left(\begin{matrix}8 \\ 7\end{matrix}\right) = 8$

#### Explanation:

When I have a sum of the sort $\left(\begin{matrix}n \\ k\end{matrix}\right) ,$ 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 $k = 0$.

$\left(\begin{matrix}n \\ k\end{matrix}\right)$ 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 $\left(\begin{matrix}n \\ n - 1\end{matrix}\right) = n$.

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, $8$.