Chapter 11.7, Problem 18E

To determine

To calculate: The number of combinations.

Answer to Problem 18E

The number of combinations are $330$ .

Explanation of Solution

Given information:

The following combination:

  $11C7$

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]

  $\Rightarrow 11C7=\frac{11P7}{7!}$   [Use combination formula]

  $=\frac{11\cdot 10\cdot 9\cdot 8\cdot 7\cdot 6\cdot 5}{7\cdot 6\cdot 5\cdot 4\cdot 3\cdot 2\cdot 1}$   [Write $11P7$ and $7!$ as a product]

  $=\frac{11\cdot 10\cdot \cancel{\stackrel{3}{9}}\cdot \cancel{\stackrel{\cancel{2}}{8}}\cdot \cancel{7}\cdot \cancel{6}\cdot \cancel{5}}{\cancel{7}\cdot \cancel{6}\cdot \cancel{5}\cdot \cancel{\underset{1}{4}}\cdot \cancel{\underset{1}{3}}\cdot \cancel{\underset{1}{2}}\cdot 1}$   [Divide out common factors]

  $=330$   [Simplify]

Therefore, the number of combinations are $330$ .