Do the following trick. Let's represent the relation between letters and envelops in this way
L1 --------- E1
L2 ---------- E2
L3 ---------- E3
L4 ---------- E4
If I want to know the possible combinations I have to permute the envelops, for instance another combination would be
L1 --------- E2
L2 ---------- E1
L3 ---------- E3
L4 ---------- E4
So the total possible combinations are the permutations of the envelops 4!
Now we want the combinations that only have one match correct.
Let's say that the first match is correct
L1 ---------- E1
now we have to generate the other combinations but without a match
L2 ---------- E2
L3 ---------- E3
L4 ---------- E4
For L2 we have two possible envelops E3 and E4.
if you pick E3, for E3 you have to pick E4 since you cannot pick E4 for L4.
if you pick E4, for E3 you have to pick E2 since you cannot pick E3 for L3.
So given the choice E3 or E4 the others are automatic. You only have two options.
Since we assume that the first match is correct but this can happen in any of the other
we have 4 x 2 = 8 combinations in which only one letter is correctly addressed.
Probability = desire outcomes/ total = 8/4! = 1/3
