What are the odds of spinning matching items in a slot machine? Lets say we have a slot machine with $5$ reels. Each reel has $5$ different items on it.

What are the odds of spinning $2, 3, 4$ and $5$ matching items?

As I understand the probability of rolling a particular item in each reel is $\frac15$. I just don't know how to calculate the combined probability of these events. Is it simply $\frac15\cdot\frac15$ for two matching items or are things more complicated?
Yes but you need to multiply $1/25$ with $\binom 5r, r=\{2,3,4,5\}$ as you need to consider permutations of an arrangements on 5 distinct reels and hence consider choosing any two reels to be possible in multiple ways, i.e. $\binom 5r$
It's a bit more complicated.

Let's first answer the question what is the probability that there are $5$ matching items


The number of all possible spins is $5^5$.
The number of good spins is $5$.
Therefore, the probability of a good spin is $\frac{5}{5^5}=5^{-4} = 0.16\%$




Now, what about four matching items?

The set of all possible spins is now bigger. Let's say that each reel can spin $1,2,3,4$ or $5$. Now, how can we get $4$ same items?

Well, we can have four ones in the first four reels, and one non-one in the fifth reel. There are $4$ such results ($11112,11113,11114,11115$). Then, there are four results for four twos, and four for four threes and so on. So, there are $5\cdot 4=20$ possibilities for having $4$ equal items in the first four reels.

You then also have $20$ possibilities for $4$ equal items on reels $1,2,3,5$, and so on, with a total of $5\cdot 20=100$ good possibilities.

Therefore, the probability of a good spin is $\frac{100}{5^5} = 3.2\%$



OK, three matching items. Well:


there are ${5\choose 2}=10$ possible triplets of reels on which you 
For each such triplet, you have $5$ possibilities for which item will appear on all three. You can then freely choose the other $2$ reels to be anything but the chosen item. Therefore, for each triplet, you have $5\cdot 4\cdot 4$ good spins.


So, $10$ possible triplets and $80$ good rolls for each triplet yields $800$ good spins. With $5^5$ total spins, the probability is
$$\frac{800}{5^5} = 25.6\%$$



What about the probability of two matching items? Well, that depends:


Do you want the probability that there is only one matching pair?
Do you allow for two pairs?
Do you allow one pair and one triple?