PERMUTATION AND COMBINATION

Permutations :

The ways of arranging or selecting smaller or equal number of persons or objects from a group of persons or collection of objects with due regard being paid to the order of arrangement or selection are called permutations.

Combinations :

The number of ways in which smaller or equal number of things are arranged or selected from a collection of things where the order of selection or arrangement is not important are called combinations.

Difference between Permutations and Combinations

Formulas

Permutations :

nPr = n!/(n - r)!

Combinations :

 nCr = n!/r!(n - r)!

Circular Permutations :

Case (i) :

Both clockwise and anti clockwise rotations are considered. (Hint : Every person has the same two neighbors) Then, the formula for circular permutations is

(n - 1)!

Case (ii) :

Either clockwise or anti clockwise rotation is considered, not both. (Hint: No person has the same two neighbors) Then, the formula for circular permutations is

(n-1)!/2

Shortcuts

1. nPr = n(n - 1)(n - 2) .... to r terms.

    Example : 7P3 = 7 x 6 x 5 = 210

2. nCr = [n(n - 1)(n - 2) ... to r terms]/r!

Example : 7P3 = [7 x 6 x 5]/[3 x 2 x 1] = 35

3. nCr = nCn-r

(we will use this property only when we  want to reduce the value of r)

Example : 25P22 = 25P3

4. nPr = r! ⋅ nCr

5. nP1 = n

6. nC1 = n

7. nP0 = 1

8. nC0 = 1

9. nPn = n!

(No. of permutations of n things taken all at a time)

10. nCn = 1

(Explanation : nCn = nCn-n = nC0 = 1)

11. Number of Permutations of n things taken all at a time, when two particular things always come together is

= (n - 1)!.2!

12. Number of Permutations of n things taken all at a time, when two particular things always do not come together is  

= n!-(n - 1)!.2!

13. The value of 0! = 1.

Fundamental Principles of Counting

Multiplication Rule :

There are two things, one can be done in m number of ways and the second can be done in n number of ways, Then the total number of ways of doing both the things is

= m x n

This rule is called multiplication rule.

AND ===> Multiplication

Addition Rule :

There are two things, one can be done in m number of ways and the second can be done in n number of ways, Then the total number of ways of doing either the first one or second one, not both is

= m + n

This rule is called addition rule.

OR ===> Addition

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