Question

In how many different ways can five elements be selected in order from a set with three elements when repetition is allowed?

Short answer

There are 243 ways in which five elements can be selected in order from a set with three elements when repetition is allowed.

Step-by-step solution

Step 1:

Definitions

Definition of Permutation (Order is important)

No repetition allowed:$P(n,r)=\frac{n!}{(n-r)}$

Repetition allowed:$n^r$

Definition of combination (order is important)

No repetition allowed:$C(n,r)=\left(\begin{array}{l}n\\ r\end{array}\right)=\frac{n!}{r!(n-r)!}$

Repetition allowed:$C(n+r-1,r)=\left(\begin{array}{c}n+r-1\\ r\end{array}\right)=\frac{(n+r-1)!}{r!(n-1)!}$

withdata-custom-editor="chemistry" $\mathrm{n}!=\mathrm{n}(\mathrm{n}-1)\cdot .....\cdot 2\cdot 1$

Step 2: Solution

The order of the elements matters (since we want to select the elements in order), thus we need to use the definition of permutation.

We are interested in selecting r = 5 elements from a set with n = 3 elements.

Repetition of elements is allowed.

$n^r=3^5=243$

Thus there are 243 ways in which five elements can be selected in order from a set with three elements when repetition is allowed.