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Found in: Page 432

### Discrete Mathematics and its Applications

Book edition 7th
Author(s) Kenneth H. Rosen
Pages 808 pages
ISBN 9780073383095

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

There are 3125 ways in which five elements can be selected in order from a set with five elements when repetition is allowed.

See the step by step solution

## Step 1: Definitions

Definition of Permutation (Order is important)

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

Repetition allowed: ${n}^{r}$

Definition of combination (order is important)

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

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

with $n!=n\left(n-1\right)\cdot .....\cdot 2\cdot 1$

## 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 = 5 elements.

Repetition of elements is allowed.

${n}^{T}={5}^{5}=3125$

Thus there are 3125 ways in which five elements can be selected in order from a set with five elements when repetition is allowed.