How many ways can a captain and a vice-captain be selected from a team of?

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Permutation and Combination

Permutation and Combination are two separate ways to represent a group of elements. Both are different and many students get confused between the two. When the order of arrangement doesn’t matter then we call it a combination. If the order does matter then we have a permutation. It can be rightly said that a permutation is an ordered combination.

1. Permutation of n different objects

If n is a positive integer and r is a whole number such that r<n, then P(n, r) represents the number of all possible arrangements or permutations of n distinct objects taken r at a time. In the case of permutation without repetition, the number of available choices will be reduced each time. It can also be represented as nPr.

P(n, r)=n(n-1)(n-2)(n-3)……..up to ‘r’ factors

⇒P(n, r)=n(n-1)(n-2)(n-3)……..(n–r+1)

⇒nPr=n!n−r!

Here, “nPr” represents the “n” objects to be selected from “r” objects without repetition, in which the order matters.

2. Permutation When Repetition is Allowed

We can easily calculate the permutation with repetition. The permutation with repetition of objects can be written using the exponent form.

When the number of objects is “n,” and we have “r” to be the selection of an object, then;

Choosing an object can be in n different ways (each time).

Thus, the permutation of objects when repetition is allowed will be equal to,

n×n×n× ……(r times) =nr

3. Permutation When the Objects are not Distinct

Permutation of n different objects when p1 objects among ‘n’ objects are similar, p2 objects of the second kind are similar, p3 objects of the third kind are similar ……… and so on, pk objects of the kth kind are similar and the remaining of all are of a different kind,

Thus it forms a multiset, where the permutation is given as:

=n!(p1! p2! p3!....... pk)

Combination Formula

1. The combination is defined as the selection of r things that can be done out of total n things. This is denoted by nCr which is equal to n!r!(n-r)

2. As per the Fundamental Principle of Counting, if a particular thing can be done in m ways and another thing can be done in n ways, then either one of the two can be done in m+n ways and both of them can be done in m×n ways.

Relation between Permutation and Combination

nPr=nCr.r!

Difference Between Permutation and Combination

Permutation	Combination
Arranging people, digits, numbers, alphabets, letters, and colours	Selection of menu, food, clothes, subjects, team.
Picking a team captain, pitcher and shortstop from a group.	Picking three team members from a group.
Picking two favourite colours, in order, from a colour brochure.	Picking two colours from a colour brochure.
Picking first, second and third place winners.	Picking three winners.

Relation Between Permutations and Combinations.

The formula for permutations and combinations are related as:

nCr=nPrr!

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