Chapter 11.7, Problem 11E

To determine

To calculate: The number of combinations.

Answer to Problem 11E

The number of combinations are $3$ .

Explanation of Solution

Given information:

The following combination:

  $3C2$

Calculation:

To find the number of combinations of $n$ objects taken $r$ at a time, divide the number of permutations of $n$ objects taken $r$ at a time by $r!$

  $nCr=\frac{n\mathrm{Pr}}{r!}$   [Combination formula]

  $\Rightarrow 3C2=\frac{3P2}{2!}$   [Use combination formula]

  $=\frac{3\cdot 2}{2\cdot 1}$   [Write $3P2$ and $2!$ as a product]

  $=\frac{3\cdot \cancel{2}}{\cancel{2}\cdot 1}$   [Divide out common factors]

  $=3$   [Simplify]

Therefore, the number of combinations are $3$ .