First you should ask yourself: "How many ways are there to choose five objects from 25?" You have 25 to choose from for your first object. Then you have 24 left from which to choose your second object. Then you have 23 left from which to choose your third object. Then you have 22 left from which to choose your fourth object. Then you have 21 left from which to choose your fifth object. That means there are $25\times 24 \times 23 \times 22 \times 21$ ways of choosing five objects from 25.
But wait! You said that the order doesn't matter. You need to ask yourself "How many ways could I have picked out those five objects?" This is the same as asking: "How many ways can I rearrange five objects?" You have five objects from which to choose your first, four from which to choose your second, three from which to choose your third, two from which to choose your fourth and one from which to choose your fifth. That means there are $5\times 4 \times 3 \times 2 \times 1$ ways of rearranging five objects. Putting all of this together, the answer you're looking for is:
$ \frac{25\times 24 \times 23 \times 22 \times 21}{5\times 4 \times 3 \times 2 \times 1} = 53,130 \, .$
In general, if you want to choose $r$ objects from a pool of $n$, and the order doesn't matter, there are $n$-choose-$r$ ways of doing that:
$^nC_r = \frac{n!}{r!(n-r)!} \, $
where "$n$-factorial" is $n! = n \times (n-1) \times (n-2) \times \cdots \times 2 \times 1.$