1. permutations with repeated objects?

How many ‘5-objected-groups’ can be formed from 22 different objects, if repeating objects can be used?

i.e. objects 1, 1, 1, 2, 1 is different from objects 2, 1, 1, 1, 1

2. There is no way to give an answer based on so little information.
You need to provide much more detail and examples.

The string “11112” can be arranged in five different ways.
The string “11122” can be arranged in $=\frac{5!}{(3!)(2)}$ different ways.
Is that close to what you mean?

3. actual question

Actual question is…

How many pentapeptides is it possible to construct using all of the 22 common amino acids if repeat units are allowed? i.e. the same amino acid can be used more than once

n.b. pentapeptide is a combination of 5 amino acids.

I thought I had to used the nPr button on the calculator, so 22P5 but I think it’s wrong

Now I think it might be 22^5

???

4. Variations with repetition

You are right. This is called Variation with repetition and is written as:

V~=n^k where n is number of "objects" and k is the number of positions.

Pn are called permutations without repetitions and are special case of variations without repetitions when n=k