Chapter 6, Problem 13T

Solution

To State:

The number of combinations of city council members that could have voted in favor of the budget increase.

  $6435$

Given:

  $8$ out of the $15$ members voted in favor of the budget increase.

Concepts Used:

The number of combinations of $n$ distinct objects taking $r$ at a time is ${}_n{C}_r=\frac{n!}{r!\left(n-r\right)!}$ .

Calculations:

Calculate the number of combinations of $8$ city council members out of total $15$ that could have voted in favor of the budget increase.

  $\begin{array}{c}{}_{15}{C}_8=\frac{15!}{8!\left(15-8\right)!}\\ =\frac{15\cdot 14\cdot 13\cdot 12\cdot 11\cdot 10\cdot 9}{7\cdot 6\cdot 5\cdot 4\cdot 3\cdot 2\cdot 1}\\ =15\cdot 13\cdot 11\cdot 3\\ =6435\end{array}$

Conclusion:

  $6435$ is the number of combinations of city council members that could have voted in favor of the budget increase.