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

Permutations

Selection is made. Beyond selection, order or arrangement is important.

Combinations

Selection is made. But arrangement or order is not important

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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