Question

A man and a woman meet at a restaurant. After a long conversation, they agree to have dinner the next day if the man remembers to call the woman to confirm the date. The next morning the man discovers that he can remember the digits in her number (1, 2, 3, 4, 5, 6, and 7), but he has completely forgotten their order. If he decides to arrange the seven digits in random order and dial every combination, what are the chances that any given number will be her phone number?

Answer 1

The chance that any given randomly dialed sequence of the seven digits is the correct phone number is 1 in 5,040, as there are 5,040 distinct permutations for seven unique digits.

Step-by-step explanation:

The chances that any given number he dials will be her correct phone number is calculated by finding the probability of arranging 7 distinct digits in a specific order. Since each of the 7 digits can only be used once, the man has 7 options for the first digit, 6 for the second, continuing down to 1 option for the seventh digit. This is a permutation problem and is solved by calculating 7 factorial (7!), which is the product of all positive integers up to 7: 7 × 6 × 5 × 4 × 3 × 2 × 1 = 5,040 distinct combinations.

Therefore, the probability that the man dials the correct number on any given try is 1 in 5,040 or 0.0001984 (which can also be expressed as roughly 0.01984%).