Final answer:
The probability that no letters are repeated when typing 10 letters is calculated using permutations where no letter can be repeated, resulting in 26!/(26-10)! divided by 26^10.
Step-by-step explanation:
To determine the probability that no letters are repeated when 10 letters are typed, we must consider the total number of letters available, which is 26 (assuming the English alphabet). Since no letters can be repeated, the first letter can be any of the 26. For the second letter, there are only 25 options left (since one letter has already been used), for the third letter 24 options, and so forth, down to 17 options for the tenth letter.
To find the total number of ways to type 10 different letters out of 26 with no repetition, we use the formula for permutations of n items taken r at a time, which is nPr = n! / (n-r)!. In this case, n = 26 and r = 10, so we get 26P10 = 26! / (26-10)! = 26! / 16!
The probability is calculated as the number of favorable outcomes (the permutations calculated) over the total outcomes possible (which, if repetition were allowed, would be 26^10 since each letter can be any of the 26). Since the no-repetition scenario limits our options, the probability will be 26P10 / 26^10.
To calculate this exactly requires working through the factorial calculations and then dividing, which generally requires a calculator for numbers as large as 26!.