Chapter 11.7, Problem 15E

To determine

To calculate: The number of combinations.

Answer to Problem 15E

The number of combinations are $4$ .

Explanation of Solution

Given information:

The following combination:

  $4C3$

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]

  $\Rightarrow 4C3=\frac{4P3}{3!}$   [Use combination formula]

  $=\frac{4\cdot 3\cdot 2}{3\cdot 2\cdot 1}$   [Write $4P3$ and $3!$ as a product]

  $=\frac{4\cdot \cancel{3}\cdot \cancel{2}}{\cancel{3}\cdot \cancel{2}\cdot 1}$   [Divide out common factors]

  $=4$   [Simplify]

Therefore, the number of combinations are $4$ .