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Question: Permutation and combination? In your own words, explain the difference between a permutation and a c...

Permutation and combination? In your own words, explain the difference between a permutation and a combination. Also, give an example of a problem whose solution is a permutation and an example of a problem whose solution is a combination?

Explanation

Solution

Permutation is defined as the arrangement of objects in a definite order whereas combination is defined as the selection of some or all objects from a given set of different objects when order of selection is not considered.

Complete step by step answer:
Permutation and combination are the methods of counting which help us to determine the number of different ways of arranging and selecting objects out of a given number of objects, without actually listing them.Permutation is defined as the arrangement of objects in a definite order. For example: we have two books, one book of each subject, English and Mathematics. We have to arrange them in a shelf so there are two ways to arrange them either we can keep English book first and then mathematics or Mathematics book first and then English book.

The number of permutations of nn different objects taken rr objects out of them without replacement is given by the following formula:
nPr=n!(n−r)!^n{P_r} = \dfrac{{n!}}{{(n - r)!}}
For above example, the number of permutations of 22 different books, English and Mathematics, taken both of them is given as:
⇒2P2=2!(2−2)!^{ \Rightarrow 2}{P_2} = \dfrac{{2!}}{{(2 - 2)!}} =2!0!=2! = \dfrac{{2!}}{{0!}} = 2!
⇒2×1=2\Rightarrow 2 \times 1 = 2
So, there are two ways to arrange these books.

Combination is defined as the selection of some or all objects from a given set of different objects where order of selection is not considered. For example: suppose we have three friends, Akhil, Nikhil and Virat, we want two of them to sit together. So, the possible number of pairs will be:
-Akhil and Nikhil
-Nikhil and Virat
-Akhil and Virat
Therefore, there are three ways to select two friends out of three friends.The number of combinations of nn different objects taken rr objects out of them without replacement is given by the following formula:
nCr=n!r!(n−r)!=nPr⁡r!^n{C_r} = \dfrac{{n!}}{{r!(n - r)!}} = \dfrac{{_n\Pr }}{{r!}}

From above example, the number of ways of picking the names of two friends out of three friends is given as:
⇒3C2=3!2!(3−2)!=3!2!=3×2×12×1=3{ \Rightarrow ^3}{C_2} = \dfrac{{3!}}{{2!(3 - 2)!}} = \dfrac{{3!}}{{2!}} = \dfrac{{3 \times 2 \times 1}}{{2 \times 1}} = 3
So, there are three ways to select two friends out of three friends. Hence, the difference between permutation and combination is that permutation is defined as the number of ways to arrange objects and combination is defined as the number of ways to select objects. In permutation order is to be considered and in combination we do not consider order. Permutation is denoted by nPr=n!(n−r)!^n{P_r} = \dfrac{{n!}}{{(n - r)!}} whereas combination is denoted by nCr=n!r!(n−r)!=nPr⁡r!^n{C_r} = \dfrac{{n!}}{{r!(n - r)!}} = \dfrac{{_n\Pr }}{{r!}}.

Note: Note that if in a problem we are asked for selection and their ordering, then we will use permutation. So, permutation== selection++ordering and if we are asked only for selection then we will use combination. So, combination== selection.