Four letters to different insureds are prepared along with accompanying envelopes. the letters are put into the envelopes randomly. calculate the probability that at least one letter ends up in its accompanying envelope.

This is a problem in probability and statistics. Questions about combinations and permutations are very tricky. You can answer them easily by using specific techniques for specific type of questions. For this type, the applicable technique is the fundamental counting principle. This is how it's applied. 

We have 4 pairs of letters and envelopes and the pairing must not be repeated. So, you count 4 pairs by multiplying 4 numbers. These numbers represent the letters. In the 1st envelope, you can put any of the 4 letters. In the 2nd envelope, you can put any of the 3 remaining letters. In the 3rd envelope, you put any of the remaining 2 and so on and so forth. So, the number of ways of arranging these 4 pairs is 4×3×2×1 = 24 ways.

To get the probability, you place the total of ways as the denominator. What you place in the numerator is the successful events. That is, at least one pair is correct. Hence, it covers 1, 2, 3 and 4 pairs. Then, you add their probabilities.
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Total Probability = 1/24 + 2/24 + 3/24 + 4/24 = 5/12 or 0.417 or 41.7%