# Overview

## Fundamental Principles Of Counting

Permutation and Combination are basically methods that enable us to count faster.

They are based on two counting principles:

• Fundamental Principle Of Multiplication

### Fundamental Principle Of Multiplication

Fundamental Principle Of Multiplication - If there are two jobs such that one of them can be completed in m ways and when it has been completed (in any of the m ways), second job can be completed in n ways; then the two jobs can be completed in m × n ways. Notice that the completion of the second job is dependent on the completion of the first job.
The keyword is AND – We need to finish the first job AND the second job after that.

Fundamental principle of multiplication is used in Permutation, as well as Combination.
##### Q. There are 2 roads connecting cities A and B and 3 roads connecting cities B and C. In how many ways can one travel from city A to C?
(a)  8        (b) 6        (c) 12        (d) 14 Explanation:

The person needs to travel to city B AND then city C

Number of ways to go from city A to city B = 2
Number of ways to go from city B to city C = 3

So, total number of ways in which one can go from city A to city C = 2 × 3 = 6 ways

##### Q. A teacher has to select 1 boy and 1 girl out of 5 boys and 6 girls. In how many ways can this be done?
(a)  30        (b) 36        (c) 28        (d) 29

Explanation:

The teacher has to select 1 boy AND 1 girl.

Number of ways to select 1 boy = 5
Number of ways to select 1 girl = 6

So, total number of ways in which the required selection can be made = 5 × 6 = 30 ways

Fundamental Principle Of Addition - If there are two jobs such that they can be performed independently in m and n ways respectively, then either of the two jobs can be performed in (m + n) ways. Notice that the completion of each of the two jobs is independent of the completion of the other job.

The keyword is OR – We need to finish the first job OR the second job.

Fundamental principle of addition is used in Combination.
##### Q. There are 2 roads connecting cities A and B and 3 roads connecting cities A and C. In how many ways can one travel from city A to city B or C?
(a)  8        (b) 10        (c) 5        (d) 7 Explanation:

The person can go to city B OR city C

Number of ways to go from city A to city B = 2
Number of ways to go from city A to city C = 3

So, total number of ways in which one can go from city A to city B or C = 2 + 3 = 5 ways

##### Q. A teacher has to select 1 boy or 1 girl out of 5 boys and 6 girls. In how many ways can this be done?
(a)  30        (b) 36        (c) 10        (d) 11

Explanation:

The teacher has to select 1 boy OR 1 girl.

Number of ways to select 1 boy = 5
Number of ways to select 1 girl = 6

So, total number of ways in which the required selection can be made = 5 + 6 = 11 ways

Basically, the teacher has to select 1 student out of 11 students. So, this can be done in 11 ways.

##### Q. The figure shows the various one-way routes between the four cities A, B, C and D. How many routes are possible to travel from the city A to city C? (a)  7       (b) 8        (c) 12        (d) 11

Explanation:

Number of ways to go from A to C = Number of ways to go from A to B to C + Number of ways to go from A to D to C = 3 × 1 + 2 × 2 = 3 + 4 = 7 ways

##### Q. There are 6 men and 9 women in an office. Find out:
1. In how many ways can a person be selected?
2. In how many ways can a team of one man and one woman be selected?

Explanation:

The number of ways in which a man can be selected = 6
The number of ways in which a woman can be selected = 9

So, number of ways in which a Man OR a Woman can to be selected = 6 + 9
(remember with OR we will do addition)

And number of ways in which a Man AND a Woman can be selected = 6 × 9 = 54
(remember with AND we will do multiplication)

## Concept of Factorial

The notation n! or ∟n represents the product of first n natural numbers (i.e. product of all natural numbers from 1 to n).

n! = 1 × 2 × 3 × . . . × (n – 1) × n

n! is called ‘n factorial’.

1 ! = 1
2 ! = 1 × 2
3 ! = 1 × 2 × 3
4 ! = 1 × 2 × 3 × 4 and so on.

Note that, 0 ! = 1

### Recursive factorial formula

We can easily calculate a factorial from the previous one, i.e. if we know (n - 1)!, we can easily calculate n!.

n! = n × (n−1)! (Recursive factorial formula)

We know that, 4 ! = 1 × 2 × 3 × 4 = 24
So, 5! = 5 × 4! = 5 × 24 = 120
6! = 6 × 5! = 6 × 120 = 720

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