Question

A rug manufacturer has decided to use 7 compatible colors in her rugs. however, in weaving a rug, only 5 spindles can be used. in advertising, the rug manufacturer wants to indicate the number of different color groupings for sale. how many color groupings using the seven colors taken five at a time are there? (this assumes that 5 different colors will go into each rug, i.e., there are no repetitions of color.)

Answer 1

The number of different color groupings for sale using the seven colors taken five at a time is $\boxed{21}.$

Further explanation:

The formula of permutation can be expressed as,

$\boxed{^n{C_r}=\dfrac{{n!}}{{r!\left( {n - r} \right)!}}}$

Here, “$n$” represents the total observations and “$r$” represents number of the observation has to be arranged.

Given:

Total number of colors is 7 and the only 5 spindles can be used.

Calculation:

The number of different color groupings for sale using the seven colors taken five at a time can be calculated as follows.

$^n{C_r} =\dfrac{{n!}}{{r!\left( {n - r} \right)!}}$

Substitute 7 for $n$ and 5 for $r$ in above equation.

$\begin{aligned}^7{C_5}&= \frac{{7!}}{{5!\left( {7 - 5} \right)!}}\\&= \frac{{7!}}{{5! \times 2!}}\\&= \frac{{7 \times 6 \times 5!}}{{5! \times 2!}}\\&= \frac{{42}}{2}\\&= 21\\\end{aligned}$

The number of different color groupings for sale using the seven colors taken five at a time is $\boxed{21}.$

Learn more:

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2. Learn more about range and domain of the function brainly.com/question/3412497

3. Learn more about profit and loss brainly.com/question/2479097

Answer details:

Grade: High School

Subject: Mathematics

Chapter: Permutation and Combination

Keywords: Different ways, select, combination, randomly picks, permutation, Rug manufacturer, 7 compatible colors, weaving a rug, 5 spindles, advertising, indicate, different colors, seven colors, no repetition of color.

Answer 2

The number of available colors is n = 7.
The number of colors that can be selected at a time among the 7 colors is r = 5.

This means that the number of selections of 5 colors from 7 colors at a time is ₇C₅.
By definition,
$_{7} C_{5} = \frac{7!}{2!5!} = \frac{7.6.5!}{2.1.5!} = \frac{7.6}{2}=21$

Answer: The number of color groupings is 21.