Chapter 9, Problem 79RE

To determine

To evaluate: $\left(\begin{array}{c}7\\ 2\end{array}\right)$ by using Pascal’s Triangle.

Answer to Problem 79RE

  $21$

Explanation of Solution

Concept Used:

Pascal’s triangle:

The n^th row of Pascal’s triangle gives the coefficients of ${\left(x+y\right)}^n$ .

Consider the given expression $\left(\begin{array}{c}7\\ 2\end{array}\right)$

This can be written as $C{}_2$

Now, in the Pascal’s triangle $C{}_r$ represents the r^th element of the n^th row of the Pascal’s triangle for $r=0,1,2,\mathrm{...},n$ .

So, $C{}_2$ is the 2^nd element of the 7^th row of the Pascal’s triangle.

The binomial coefficients from the seventh row of Pascal’s Triangle are 1, 7, 21, 35, 35, 21, 7, 1.

The second element of this row is 21.

Thus, $\left(\begin{array}{c}7\\ 2\end{array}\right)=21$