Chapter 6.1, Problem 38E

Solution

To find the number of ways the students can be chosen for the two positions.

The total number of ways in which two students can be chosen in 132.

Given information:

There 12 students and two position.

Formula used:

  $C{}_r=\frac{n!}{\left(n-r\right)!\cdot r!}$

Calculation:

One student out 12 can be chosen for first position in $C{}_1$ ways. One student out of the remaining 11 student can be chosen for the second position in $C{}_1$.

So, the total number of ways in which two position can be filled is $C{}_1\times C{}_1$.

Simplify $C{}_1\times C{}_1$.

  $\begin{array}{c}C{}_1\times C{}_1=\frac{12!}{\left(12-1\right)!\cdot 1!}\times \frac{11!}{\left(11-1\right)!\cdot 1!}\\ =\frac{12!}{11!\cdot 1!}\times \frac{11!}{10!\cdot 1!}\\ =12\times 11\\ =132\end{array}$

Conclusion:

The total number of ways in which two students can be chosen in 132.