Question

a) What is the difference between an r-combination and an r-permutation of a set with n elements?

b) Derive an equation that relates the number of r-combinations and the number of r-permutations of a set with n elements.

c) How many ways are there to select six students from a class of 25 to serve on a committee?

d) How many ways are there to select six students from a class of 25 to hold six different executive positions on a committee?

Short answer

(a) The difference between the quantities states in the question is that in an r - permutation the order of r element matters while a r - combination the order of r - elements do not matter.

(b) The number of r - permutation and the number of r - combination of a set with n elements are related as follows.

$\mathrm{P}(\mathrm{n},\mathrm{r})=\mathrm{C}(\mathrm{n},\mathrm{r})\times \mathrm{r}!$

(c) There are 177100 numbers of ways to select six students from a class of 25 to serve on a committee.

(d) The number of ways to select six students from a class of 25 to hold six different executive positions$=1,27,512,000$ .

Step-by-step solution

Definition of Concept

Functions: It is a expression, rule or law which defines a relationship between one variable and another variables.

Find difference between an r-combination and an r-permutation of a set with n elements

(a)

Considering the given information:

r Combination and r- permutation of a set with n elements.

Using the following concept:

Permutation is the act of altering the arrangement of a group of items, particularly their linear order.

A r-permutation specifies the order and selection of r elements, whereas a r-combination does not.

The order of r-elements is important in a r-permutation, but it is not important in a r-combination.

Therefore, the difference between the quantities states in the question is that in a r - permutation the order of r element matters while a r - combination the order of r - elements do not matter.

Find the equation

(b)

Considering the given information:

Number of permutation =r

Number of element =n

Using the following concept:

The number of r - permutation of a set with n elements is given by

${\mathrm{n}}_{{\mathrm{p}}_{\mathrm{r}}}=\frac{\mathrm{n}!}{\mathrm{nr}!}......\left(1\right)$

The number of r - combination of a set with n elements is given by

${\mathrm{n}}_{{\mathrm{c}}_{\mathrm{r}}}=\frac{\mathrm{n}!}{\mathrm{r}!\mathrm{n}-\mathrm{r}!}......\left(2\right)$

To find the relationship between ${\mathrm{n}}_{{\mathrm{c}}_{\mathrm{r}}}$and ${\mathrm{n}}_{{\mathrm{p}}_{\mathrm{r}}}$compare the two equations.

$$\begin{gathered} {\mathrm{n}}_{{\mathrm{p}}_r}=\frac{\mathrm{n}!}{\mathrm{n}-r!} \\ {\mathrm{n}}_{{\mathrm{c}}_r}=\frac{\mathrm{n}!}{\mathrm{r}!\mathrm{n}-r!} \\ {\mathrm{n}}_{{\mathrm{c}}_r}=\frac{{\mathrm{n}}_{{\mathrm{p}}_r}}{\mathrm{r}!} \\ {\mathrm{n}}_{{\mathrm{p}}_r}=r!{\mathrm{n}}_{{\mathrm{c}}_r} \end{gathered}$$

Therefore, the required number of r - permutation and the number of r - combination of a set with n elements are ${\mathrm{n}}_{{\mathrm{p}}_r}=r!{\mathrm{n}}_{{\mathrm{c}}_r}$.

Find the number of ways to select six student from a class of 25 to serve on a committee

(c)

Considering the given information:

Number of student =6

Number of class =25

Using the following concept:

Number of ways to select n students from a class of r to serve on a committee${\mathrm{n}}_{{\mathrm{c}}_r}$

As a result, the number of ways to choose six students from a class of 25 to serve on a committee equals the number of six combinations of a set of 25 elements.

$$\begin{gathered} C(25,6)=\frac{25!}{6!19!} \\ =\frac{25\times 24\times 23\times 22\times 21\times 20\times 19!}{6!19!} \\ =\frac{25\times 24\times 23\times 22\times 21\times 20}{6\times 4\times 3\times 2} \\ =177100 \end{gathered}$$

Therefore, there are 177100 numbers of ways to select six students from a class of 25 to serve on a committee.

Find the number of ways to select six students from a class of 25 to hold six different executive positions

(d)

Considering the given information:

Number of student =6

Number of class=25

Number of different executive position =6

Using the following concept:

Number of ways to select n students from a class of r to serve on a committee is${\mathrm{n}}_{{\mathrm{c}}_{\mathrm{r}}}$.

As a result, the number of ways to choose six students from a class of 25 to hold six different executive positions on a committee equals the number of 6 - permutations of a set of 25 elements.

Mathematically,

Number of ways

$$\begin{gathered} P(25,6)=\frac{25!}{19!} \\ =\frac{25\times 24\times 23\times 22\times 21\times 20\times 19!}{19!} \\ =1,27,512,000 \end{gathered}$$

Therefore, the required number of ways to select six students from a class of 25 to hold six different executive positions is $1,27,512,000$.