# Permutations

Each of the arrangements, which can be made by taking, some or all of a number of things is called a PERMUTATION.

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Result - 1 : **To find the number of permutations of 'n' things taken 'r' at a Time :

The number of ways of filling 'r' places with 'n' things

=

The above formula for ^{n}P_{r} involves following conditions :

- All the things are distinct.
- Repetition of things is not allowed in any of the arrangements.
- No arrangement is repeated.
- The arrangement is linear.

**Result**** ****- 2 : **The number of permutations of 'n' things taken all at a time.

This will be given by above formula after taking r = n.

Thus, required number of ways = ^{n}P_{n} = n!

**Result**** ****- 3 : **To find the number of permutations of 'n' things taken all at a time, when 'p' are alike of one kind, 'q' are alike of Second, 'r' alike of Third, and so on :

Let 'x' be the required number of permutations.

If p alike things are replaced by p distinct things, which are also different from others, then without changing the positions of other things these new p-things can be arranged in p! ways.

Each of 'x' permutations will give p! permutations. Thus the total number of permutations now are x (p!)

With a similar argument for 'q' - alike and 'r' - alike things, we get that if all things are different the number of permutations would be x(p!) (q!) (r!)

But number of permutations of 'n' distinct things; taken all at a time = ^{n}P_{n} = n!, thus,

**Result**** **** - 4 : **To find the number of Permutations of 'n' different things, taking 'r' at a time, when each thing can be repeated 'r' times:

In the problem we have to fill 'r' vacant places with 'n' things with repetition. Obviously, each place can be filled in 'n' ways, leaving again n ways for the other place.

Hence, the number of ways of filling r-places with 'n' things = n Ã— n Ã— n Ã— ....... Ã— n (r factors) = n^{r}

**Result**** ****- 5 : **Number of circular permutations of 'n' distinct objects :

The total number of circular permutations of 'n' distinct things is (n â€“ 1)!.

If no distinction is made between anti-clockwise and clockwise arrangements, then the number of permutations is (n â€“ 1)!