Chapter 12.6, Problem 39PPE

To determine

To calculate: The number of different groups of 5 fabrics chosen for quilt.

Answer to Problem 39PPE

There are $142,506$ different groups.

Explanation of Solution

Given information:

There are 30 fabrics available at a quilt store.

Formula used:

A combination is a selection of objects without regard to order. The expression $C{}_r$ represents the number of combination of $n$ objects chosen $r$ at a time.

  $C{}_r=\frac{n!}{r!\left(n-r\right)!}$

For any positive integer $n$, the expression $n$ factorial is written as $n!$ and is the product of integers from n down to 1.

Calculation:

According to the question,

Number of fabrics available are 30.

Number of fabrics needed are 5.

Now,

  $\begin{array}{l}C{}_5=\frac{30!}{5!\left(30-5\right)!}\\ =\frac{30!}{5!25!}\end{array}$

  $\begin{array}{l}=\frac{30\times 29\times 28\times 27\times 26\times 25\times 24\times 23\times 21\times 20\times 19\times 18\times 17\times }{25\times 24\times 23\times 22\times 21\times 20\times 19\times 18\times 17\times 16\times 15\times 14\times 13\times 12\times 11\times }\\ \frac{16\times 15\times 14\times 13\times 12\times 11\times 10\times 9\times 8\times 7\times 6\times 5\times 4\times 3\times 2\times 1}{10\times 9\times 8\times 7\times 6\times 5\times 4\times 3\times 2\times 1\times 5\times 4\times 3\times 2\times 1\times 3\times 2\times 1}\end{array}$

  $\begin{array}{l}=\frac{30\times 29\times 28\times 27\times 26}{5\times 4\times 3\times 2\times 1}\\ =6\times 29\times 7\times 9\times 13\\ =142,506\end{array}$

Hence, there are $142,506$ different groups of 5 fabrics that a person can select for quilt .