You need to think about the number of combinations that are possible and note that there arele only 10 matches (each being a 1-1 match) for this problem.
Fix 1 envelope and you have 10 possibilities, the next has 9, then 8 then so on down to 1. This is 10! possibilities for a particular sequence of envelopes.
Now we have to factor the permutations of the envelopes and this also happens to be 10! using a similar argument.
Thus the total number of envelope/card combinations is 10!*10! and since we have 10 possible outcomes we have for each outcome a probability of 1/(10!*10!) for each of the ten outcomes.
Now you need to use probability laws to get P(Card1=Envelope1 OR Card2=Envelope2 ... OR ... OR Card10=Envelope10).
The above assumes that every possible way of putting in the 10 cards in 10 envelopes is taken into account.