## Permutation and Combination

Permutation : Permutation means *arrangement* of things. The word *arrangement *is used, if the order of things *is considered*.

Combination: Combination means *selection* of things. The word *selection* is used, when the order of things has *no importance*.

**Example:** Suppose we have to form a number of consisting of three digits using the digits **1,2,3,4**, To form this number the digits have to be *arranged*. Different numbers will get formed depending upon the order in which we arrange the digits. This is an example of *Permutation*.

Now suppose that we have to make a team of 11 players out of 20 players, This is an example of *combination*, because the order of players in the team will not result in a change in the team. No matter in which order we list out the players the team will remain the same! For a different team to be formed at least one player will have to be changed.

before start topic understand Factorial

*Factorial Notation:*

Let *n* be a positive integer. Then, factorial *n*, denoted *n*! is defined as:

*n! = n(n – 1)(n – 2) … 3.2.1.*

*Examples:*

- We define
*0! = 1*. - 4! = (4 x 3 x 2 x 1) = 24.
- 5! = (5 x 4 x 3 x 2 x 1) = 120.

## Permutation:

The different arrangements of a given number of things by taking some or all at a time, are called permutations.

*Examples:*

- All permutations (or arrangements) made with the letters
*a*,*b*,*c*by taking two at a time are ().*ab*,*ba*,*ac*,*ca*,*bc*,*cb* - All permutations made with the letters
*a*,*b*,*c*taking all at a time are:

(*abc*,*acb*,*bac*,*bca*,*cab*,*cba*

**Rule/Formula**

Number of all permutations of *n* things, taken *r* at a time, is given by:

^{n}P_{r} = *n*(*n* – 1)(*n* – 2) … (*n* – *r* + 1) =*n*! **/ **(*n* – *r*)!

*Examples:*

^{6}P_{2}= (6 x 5) = 30.^{7}P_{3}= (7 x 6 x 5) = 210.*Cor. number of all permutations of**n*things, taken all at a time =*n*!.

*An Important Result:*

If there are *n* subjects of which *p*_{1} are alike of one kind; *p*_{2} are alike of another kind; *p*_{3} are alike of third kind and so on and *p*_{r} are alike of *r*^{th} kind,

such that (*p*_{1} + *p*_{2} + … *p*_{r}) = *n*.

Then, number of permutations of these *n* objects is = *n*! **/ **(*p*_{1}!).(*p*_{2})!…..(*p*_{r}!)

*Cmbinations:*

Each of the different groups or selections which can be formed by taking some or all of a number of objects is called a *combination*.

*Examples:*

- Suppose we want to select two out of three boys A, B, C. Then, possible selections are AB, BC and CA.Note: AB and BA represent the same selection.
- All the combinations formed by
*a*,*b*,*c*taking.*ab*,*bc*,*ca* - The only combination that can be formed of three letters
*a*,*b*,*c*taken all at a time is.*abc* - Various groups of 2 out of four persons A, B, C, D are:
*AB, AC, AD, BC, BD, CD*. - Note that
*ab**ba*are two different permutations but they represent the same combination.

*Number of Combinations:*

The number of all combinations of *n* things, taken *r* at a time is:

^{n}C_{n} = n! / (r!)(n-r)!

*Note:*

^{n}C_{n} = 1 and ^{n}C_{0} = 1.

^{n}C_{r} = ^{n}C_{(n – r)}

*Examples:*

^{11}C_{4}=(11 x 10 x 9 x 8)**/**(4 x 3 x 2 x 1) = 330

Problem Practice: click here