## 7. Permutations And Combinations # 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 nPr 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 nr

The number of permutations of n objects taken all at a time, where p1 objects are of first kind, p2 objects are of the second kind, ..., pk objects are of the kth 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 nCr 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

## Lecture - 5

• NCERT Miscellaneous Exercise (Q1 to Q11)