# PERMUTATIONS AND COMBINATIONS

## Summary

*Fundamental principle of counting If an event can occur in m different ways, following which another event can occur in n different ways, then the total number of occurrence of the events in the given order is m × n.*

*The number of permutations of n different things taken r at a time, where repetition is not allowed, is denoted by ^{n}P_{r} and is given by {^n}P_r = \frac{n!}{(n-r)!}, where 0 ≤ r ≤ n.*

*n! = 1 × 2 × 3 × ...×n *

*n! = n × (n – 1) !*

*The number of permutations of n different things, taken r at a time, where repetition is allowed, is n ^{r}*

*The number of permutations of n objects taken all at a time, where p _{1} objects are of first kind, p_{2} objects are of the second kind, ..., p_{k} objects are of the k^{th} kind and rest, if any, are all different is \frac{n!}{{p_1}! {p_2}! ... {p_k}!} *

*The number of permutations of n different things taken r at a time, where repetition is not allowed, is denoted by ^{n}C_{r} and is given by {^n}C_r = \frac{n!}{r! (n-r)!}, where 0 ≤ r ≤ n.*

## Lecture - 1

- Fundamental Principle of Counting
- NCERT Exercise 7.1 (Q1 to Q6)
- Factorial of a number
- NCERT Exercise 7.2 (Q1 to Q5)

## Lecture - 2

- Meaning of Permutations
- Derivation for rule of Permutations
- NCERT Exercise 7.3 (Q1 to Q9)

## Lecture - 3

- Number of Permutations when an alphabet repeat
- NCERT Example 14 and Example 16
- NCERT Exercise 7.3 (Q10 & Q11)

## Lecture - 4

- Meaning of Combination, how combination is different from permutations, derivation of rule for combinations through permutations
- Shortcuts and special cases of combinations
- NCERT Exercise 7.4 (Q1 to Q9)
- NCERT Example 19