Permutations and Combinations-2

Each of the arrangement which can be made by taking some or all of a number of things is called a permutation.



Theorem 1

Let r and n be positive integers such that 1≤r≤n. then the number of all permutations of n distinct things taken r at a time is given by

n(n-1)(n-2)…(n-(r-1))

Notation: Let r and n be positive integers such that 1≤r≤n. then the number of all permutations of n distinct things taken r at a time is denoted by the symbol P(n,r) or ⁿ C_r.

Then P(n,r) = ⁿ C_r = n(n-1)(n-2)…(n-(r-1))


Theorem 2

P(n,r) = ⁿ C_r = n!/(n-r)!

Theorem 3

The number of all permutations of n distinct things taken all at a time is n!.

Theorem 4

0! = 1


8.5 Permutations under certain conditions
Three theorems

Theorem 1
The number of all permutations of n different objects taken r at a time, when a particular object is to be always included in each arrangement is r.^n-1C_r-1

Theorem 2

The number of all permutations of n different objects taken r at a time, when a particular object is never taken in each arrangement is, ^n-1C_r-1

Theorem 3

The number of all permutations of n different objects taken r at a time, when two specified objects always occur together is 2!(r-1) ^n-2C_r-2