The given question is:
27 people are in a room each with an identical deck of poker cards. Each one draws a card from their deck.
Let X be the number of the types of card drawn. For example, if everyone draws the same card, X=1, but if everyone draws a different card, X=27. What is the expected value of X?
I have calculated the odds that at least one card of type E to be drawn is: $$1-(51/52)^27 = 0.408023$$
There is also a hint given which I cannot make any sense of:
Don't focus on whether or not each person has a unique card; that actually doesn't help you with the answer (if there are four people that share a card with someone else, does that mean all four are the same, or that they have 2 pairs?). Instead, focus on the likelihood of each type of card being chosen at least once.
Using the expected value formula:
$$ E(X) = \sum_{i=1}^{n} p_i * O_i$$
Where $O_i$ is the outcome of the $i^{th}$ event occurring and $p_i$ is the probability of that event occurring. In this case we know the expected value for the event everyone draws the same card would be: $$E(X=1)= (\frac1{52})^{27} * 1$$
I can also calculate the event that no one has the same card: $$E(X=27)= (\frac1{52})^{27} * 27$$
Where am I wrong and is there a better way approach this problem?