Chapter 9, Problem 80RE

To determine

To evaluate: $\left(\begin{array}{c}10\\ 4\end{array}\right)$ by using Pascal’s Triangle.

Answer to Problem 80RE

  $210$

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 $\left(\begin{array}{c}10\\ 4\end{array}\right)$

This can be written as $C{}_4$

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, $C{}_4$ is the 4^th element of the 10^th row of the Pascal’s triangle.

The binomial coefficients from the tenth row of Pascal’s Triangle are 1, 10, 45, 120, 210, 252, 210, 120, 45, 10, 1.

The fourth element of this row is 210.

Thus, $\left(\begin{array}{c}10\\ 4\end{array}\right)=210$