The five digits of 12345 are shuffled at random to make a new five-digit number, every digit being used once. What is the probability that the new number neither starts with 5 nor ends in 1?

There are 5! = 120 numbers. It is easier to count the bad ones. Numbers beginning with 5: 4! = 24. Numbers ending with 1: 4! = 24. Numbers doing both: 3! = 6, and these have been counted twice, so the bad numbers total 24 + 24 − 6 = 42. The good numbers are 120 − 42 = 78, a probability of 78/120 = 13/20. The value 7/20 = 42/120 is the probability of the bad event itself; 3/5 = 72/120 comes from subtracting 24 + 24 without adding the 6 double-counted numbers back; and 4/5 = 96/120 imposes only one of the two conditions.