**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-1}C

_{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-1}C

_{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-2}C

_{r-2}

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