#### Problem Of the week: 29th January 2024

#### Michael Wu

Think about how duplicates of letters may affect the answer. Also note the space and that capital letters are different to their lower case equivalents.

We can look at this problem chronologically, how many possible options does the typist have for the first character? There are 12 options in ‘Eton College’, so there are 12 options for the first character. Now we look at the second character, since we have already chosen an option for the first character, there are only 11 options. This pattern continues for the entire phrase, and can be summed up with a factoral (those unfamiliar with this term: 6! = 6x5x4x3x2x1): 12!
However, we have not accounted for any duplicate solutions; as there are 3 letters which are repeated once (e, o, l). Therefore we must divide by 2! three times, leaving us with the following as the solution:
12!/ 2! x 2! x 2! = 59,875,200.