Chapter 13.2, Problem 66AYU

Explain the difference between a permutation and a combination. Give an example to illustrate your explanation.

To determine

To find: The difference between a permutation and a combination. Give an example to illustrate your explanation.

Answer to Problem 66AYU

The difference between combinations and permutations is ordering. With permutations we care about the order of the elements, whereas with combinations we don't.

An example to illustrate :

How do we select 2 alphabets from a list of 5 alphabets?

$S=\left\{a,b,c,d,e\right\}$

$\text{Permutation }\left(\text{order matters}\right)=\left\{ab, ac, ad, ae, ba, bc, bd, be, ca, cb, cd, ce, da, db, dc, de, ea, eb, ec, ed\right\}$

$=P\left(5,2\right)=\frac{5!}{3!}=20$

$\text{Combination }\left(\text{order does not matters}\right)=\left\{ab, ac, ad, ae, bd, bc, be, cd, ce, de\right\}$

$=C\left(5,2\right)=\frac{5!}{3!2!}=10$

Explanation of Solution

Given:

Consider any permutation and combination.

Calculation:

Permutations and combinations, the various ways in which objects from a set may be selected, generally without replacement, to form subsets. This selection of subsets is called a permutation when the order of selection is a factor, a combination when order is not a factor.

Formula for permutation is $P\left(n,r\right)$.

Formula for combination is $C\left(n,r\right)$.

Difference between Permutation and combination:

The difference between combinations and permutations is ordering. With permutations we care about the order of the elements, whereas with combinations we don't.

Eg: How do we select 2 alphabets from a list of 5 alphabets?

$S=\left\{a,b,c,d,e\right\}$

Permutation (order matters) $=\left\{ab, ac, ad, ae, ba, bc, bd, be, ca, cb, cd, ce, da, db, dc, de, ea, eb, ec, ed\right\}$

$=P\left(5,2\right)=\frac{5!}{3!}=20$

Combination (order does not matters) $=\left\{ab, ac, ad, ae, bd, bc, be, cd, ce, de\right\}$

$=C\left(5,2\right)=\frac{5!}{3!2!}=10$