Chapter 13, Problem 8CT

To determine

To find: How many different color mixes are possible, assuming that no color is selected more than once.

Answer to Problem 8CT

There are $54,264$ ways to choose colors. .

Explanation of Solution

Given:There are 21colors to choose from, and customers are allowed to select up to 6 different colors.

Concept used: $C\left(n,r\right)=\frac{n!}{r!\left(n-r\right)!}$

There are 21colors to choose from, and customers are allowed to select up to 6 different colors (given).

Now it has to selectmore than once and 6 different colors are chosen.

An arrangement, without regard to order, of $r$ objects selected from $n$ distinct objects, where $r\le n$

  $\begin{array}{l}C\left(n,r\right)=\frac{n!}{r!\left(n-r\right)!}\\ C\left(21,6\right)=\frac{21!}{6!\left(21-6\right)!}\\ =\frac{21!}{15!6!}\end{array}$

  $\begin{array}{l}=\frac{21\times 20\times 19\times 18\times 17\times 16\times 15!}{\left(6\times 5\times 4\times 3\times 2\times 1\right)15!}\\ =\frac{21\times 20\times 19\times 18\times 17\times 16}{6\times 5\times 4\times 3\times 2\times 1}\\ =54,264\end{array}$

Hence, there are $54,264$ ways to choose 6 different colors from the 21 available colors.