Chapter 11.7, Problem 1E

To determine

To complete the given sentence.

Answer to Problem 1E

The complete sentence is:

You can write the number of combinations of $12\text{ objects taken 3 at a time}$ as ${}_{12}{C}_3$ .

Explanation of Solution

Given information:

Consider the sentence provided in the question:

You can write the number of combinations of $12\text{ objects taken 3 at a time}$ as __.

Concept used:

The number of combinations of $n$ objects taken $r$ at a time is represented as ${}_n{C}_r$ .

Calculation:

As per the problem,

The number of combinations of $n\text{ objects taken }r\text{ at a time}$ is written as:

  ${}_n{C}_r$

Substitute $\text{12 for }n\text{ and 3 for }r$

  ${}_{12}{C}_3$

Hence,

The number of combinations of $\text{12 objects taken 3 at a time}$ is written as ${}_{12}{C}_3$ .

Therefore, completing the given sentence:

You can write the number of combinations of $12\text{ objects taken 3 at a time}$ as ${}_{12}{C}_3$ .

Further,

Using combinations formula,

  ${}_n{C}_r=\frac{{}_n{P}_r}{r!}$

Substitute $\text{12 for }n\text{ and 3 for }r$

  ${}_{12}{C}_3=\frac{{}_{12}{P}_3}{3!}$

  $=\frac{12\cdot 11\cdot 10}{3\cdot 2\cdot 1}$ [Write ${}_{12}{P}_3$ and $3!$ as products]

  $=220$ [Divide out common factors]

  ${}_{12}{C}_3=220$ .