Chapter 8.4, Problem 71E

To determine

To evaluate: The binomial coefficient of $C{}_4$ by using Pascal’s triangle.

Answer to Problem 71E

The binomial coefficient of $C{}_4$ is $35$ .

Explanation of Solution

Given information: The expression is $C{}_4$ .

Calculation:

First write out the Pascal’s triangle such that the row that begins with 1, 7.

  $\begin{array}{ccccccccccccccc}& & & & & & & 1& & & & & & & \\ & & & & & & 1& & 1& & & & & & \\ & & & & & 1& & 2& & 1& & & & & \\ & & & & 1& & 3& & 3& & 1& & & & \\ & & & 1& & 4& & 6& & 4& & 1& & & \\ & & 1& & 5& & 10& & 10& & 5& & 1& & \\ & 1& & 6& & 15& & 20& & 15& & 6& & 1& \\ 1& & 7& & 21& & 35& & 35& & 21& & 7& & 1\end{array}$

From the Pascal’s triangle, it is observed that the number 35 in last row and the fifth column is $C{}_4$ .

That is, $C{}_4=35$ .

Thus, the binomial coefficient of $C{}_4$ is $35$ .