# How do you evaluate ((8), (5)) using Pascal's triangle?

Feb 26, 2017

$56$

#### Explanation:

"by definition "((n),(r))=(n!)/(r!(n-r)!)

((8),(5))=(8!)/(5!(8-5)!)

=(8!)/(5!xx3!)=(8xx7xx6xxcancel(5!))/(cancel(5!)xx3!)

=(8xx7xxcancel(6))/cancel(3!)

$= 56$

Feb 26, 2017

$56$

#### Explanation: $\left(\begin{matrix}8 \\ 5\end{matrix}\right) \text{ represents 8th row , 5th column}$

From Pascal's triangle the entry in the 8th row / 5th column is

$\left(\begin{matrix}8 \\ 5\end{matrix}\right) = 56$

$\text{8th row } \rightarrow 1 \textcolor{w h i t e}{x} 8 \textcolor{w h i t e}{x} 28 \textcolor{w h i t e}{x} 56 \textcolor{w h i t e}{} \textcolor{w h i t e}{x} 70 \textcolor{w h i t e}{x} \textcolor{red}{56} \textcolor{w h i t e}{x} 28 \textcolor{w h i t e}{x} 8 \textcolor{w h i t e}{x} 1$

$\textcolor{w h i t e}{\times \times \times \times \times \times \times \times \times x} \uparrow$

$\textcolor{w h i t e}{\times \times \times \times \times \times \times \times \times} \text{5th column}$