I found the following problem in the book Understanding Probability:
An ordinary deck of 52 cards is thoroughly shuffled. The dealer turns over the cards one at a time, counting as he goes ``ace, two, three, $\ldots$ , king, ace, two, $\ldots$," and so on, so that the dealer ends up calling out the thirteen ranks four times each. A match occurs if the card that comes up matches the rank called out by the dealer as he turns it over.
The author states that the inclusion-exclusion rule can be used to find the value 0.9838 for the probability of at least one match. I could not reproduce this result.
As pointed by Joriki, a derivation of this result is given in
Derangements with repetitive numbers
This quite complicated derivation is based on an integral representation involving a Laguerre polynomial of degree four. However, for the specific example, you would expect that a a direct application of the inclusion-exclusion formula must be possible.
