Probability of Letters in Envelopes: A1, A2, A3, A4, A5

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In the scenario where Mr. X's son randomly distributes letters to five recipients (A1, A2, A3, A4, A5), the probability that A1 receives his letter is 1/5. For both A1 and A2 to receive their letters, the probability is calculated as 1/20. If A1 receives his letter while A2 does not, the probability is 3/20. To determine the probability that at least one of A1, A2, or A3 receives their letter, the calculation yields 3/5. Lastly, the probability that no one receives their letter is 1/5.
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Scenario: Mr. X is writing letters to five persons A1, A2, A3, A4, A5. After Mr. X has written them he has to leave the room where the letters and envelopes are. Mr X's son, who can't read, decides to help his dad and puts each letter in different envelopes. What is the probability that:

a) A1 gets his letter
P = 1/5.

b) A1 and A2 both gets their letters
P = (1/5)(1/4) = 1/20.

c) A1 gets his letter and A2 doesn't get his
P = (1/5)(3/4) = 3/20.

d) at least one of A1, A2, A3 gets his/their letter
Need help with this one.
P = 1 - (4/5)(3/4)(2/3) = 1 - 2/5 = 3/5 ?

e) no one gets their letter
P = 4!/5! = 1/5.
 
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