Chapter 9.5, Problem 14E

To determine

To evaluate: The binomial coefficient of ${}_6{C}_3$ by using Pascal’s triangle.

Answer to Problem 14E

The binomial coefficient of ${}_6{C}_3$ is $15$ .

Explanation of Solution

Given information: The expression is ${\left(y-1\right)}^6$ .

Formula used:Binomial Theorem: ${\left(x+y\right)}^n=x^n+n{x}^{n-1}y+\cdots +nx{y}^{n-1}+y^n$ .

Calculation:

First write out the Pascal’s triangle such that the row that begins with 1, 6.

  $\begin{array}{ccccccccccccc}& & & & & & 1& & & & & & \\ & & & & & 1& & 1& & & & & \\ & & & & 1& & 2& & 1& & & & \\ & & & 1& & 3& & 3& & 1& & & \\ & & 1& & 4& & 6& & 4& & 1& & \\ & 1& & 5& & 10& & 10& & 5& & 1& \\ 1& & 6& & 15& & 20& & 15& & 6& & 1\end{array}$

From the Pascal’s triangle, it is observed that the number 15 in last row and the third column is ${}_6{C}_3$ .

That is, ${}_6{C}_3=15$ .

Thus, the binomial coefficient of ${}_6{C}_3$ is $15$ .