Chapter 11.7, Problem 5E

To determine

To calculate: The number of combinations.

Answer to Problem 5E

The number of combinations are $1$ .

Explanation of Solution

Given information:

The following combination:

  $3C3$

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 $r!$

  $nCr=\frac{nPr}{r!}$   [Combination formula]

  $\Rightarrow 3C3=\frac{3P3}{3!}$   [Use combination formula]

  $=\frac{3\cdot 2\cdot 1}{3\cdot 2\cdot 1}$   [Write $3P3$ and $3!$ as a product]

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

  $=1$   [Simplify]

Therefore, the number of combinations are $1$ .