# 7. Permutations And Combinations

“A problem is a chance for you to do your best.” – Duke Ellington

## PERMUTATIONS AND COMBINATIONS

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.