# PERMUTATION AND COMBINATION

Permutation and Combination :

In this section, we are going to learn the formulas, shortcuts and more stuff related to permutations and , combinations.

If you need practice problems on permutations and combinations,

## Permutations and Combinations - Definition

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

 PermutationsSelection is made. Beyond selection, order or arrangement is important. CombinationsSelection is made. But arrangement or order is not important

## 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

## Permutations and Combination - Shortcuts

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

Example : 7P3  =  7x6x5  =  210

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

Example : 7P3  =  [7x6x5]/[3x2x1]  =  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. No. of Permutations of n things taken all at a time, when two particular things always come together is

=  (n-1)!.2!

12.  No. 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

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. If you need practice problems on permutations and combinations,

After having gone through the stuff given above, we hope that the students would have understood permutations and combinations.

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WORD PROBLEMS

Word problems on simple equations

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OTHER TOPICS

Profit and loss shortcuts

Percentage shortcuts

Times table shortcuts

Time, speed and distance shortcuts

Ratio and proportion shortcuts

Domain and range of rational functions

Domain and range of rational functions with holes

Graphing rational functions

Graphing rational functions with holes

Converting repeating decimals in to fractions

Decimal representation of rational numbers

Finding square root using long division

L.C.M method to solve time and work problems

Translating the word problems in to algebraic expressions

Remainder when 2 power 256 is divided by 17

Remainder when 17 power 23 is divided by 16

Sum of all three digit numbers divisible by 6

Sum of all three digit numbers divisible by 7

Sum of all three digit numbers divisible by 8

Sum of all three digit numbers formed using 1, 3, 4

Sum of all three four digit numbers formed with non zero digits

Sum of all three four digit numbers formed using 0, 1, 2, 3

Sum of all three four digit numbers formed using 1, 2, 5, 6 