Question

The number of permutation of $n$ different objects taken $r$ at a time, when repetition of objects in the permutation is allowed is ${{n}^{r}}$

Answer 1

To prove the theorem we use permutation concept. Firstly we will take the number of possibilities for each object which is taken $r$ at a time when the repetition of objects is allowed. Then we will combine all the possibilities and get the desired answer.

Complete step-by-step solution:
We have that number of possibility for each object is:
$n$
Then we know we are taking different object for the given time:
$r$
We can show the above condition using diagram as follows:

So we get total permutation as,
$n\times n\times n\times n\times n......r$ Times
So we can write above value as:
${{n}^{r}}$
For example- How many three letter word can be formed by with or without meaning using the word $KIDNEY$ when repetition is allowed
So the word $KIDNEY$ has 6 words in it and we have to form 3-letter words.
So we have
$n=6$
$r=3$
Thus the permutation will be as follows:
$P$(3 letters word) $={{6}^{3}}$
$P$(3 letters word) $=216$
Hence the number of permutation of $n$ different objects taken $r$ at a time, when repetition of objects in the permutation is allowed is ${{n}^{r}}$

Note: Permutation of a set is an arrangement of its members into a sequence or linear order. Permutations are of three types: firstly we have permutation of $n$ different objects when repetition is not allowed then we have Permutation when repetition is allowed and lastly we have Permutation when the objects are not distinct or we can say Permutation of multi sets.