If $x\%$ of four-digit numbers have a repeated digit (the repeated digits do not need to be adjacent), then what is $x$? Express your answer as a decimal to the nearest tenth.
I need to find the fraction of four-digit numbers that have a repeated digit, and then multiply that by 100 to get the percentage. There are 9000 possible four-digit numbers, from 1000 to 9999. How many of them have a repeated digit? One way to approach this is to count the numbers that do not have a repeated digit, and then subtract that from 9000. To not have a repeated digit, the number must have four distinct digits. The first digit can be any of 9 choices, from 1 to 9 (it cannot be 0). The second digit can be any of the remaining 9 choices, including 0 (it cannot be the same as the first digit). The third digit can be any of the remaining 8 choices (it cannot be the same as the first or second digit). The fourth digit can be any of the remaining 7 choices (it cannot be the same as the first, second, or third digit). So, the number of four-digit numbers that do not have a repeated digit is 9 x 9 x 8 x 7. I can use a calculator to multiply these numbers: 9 x 9 x 8 x 7 = 4536. Therefore, the number of four-digit numbers that do have a repeated digit is 9000 - 4536 = 4464. The fraction of four-digit numbers that have a repeated digit is 4464/9000. To get the percentage, I multiply by 100 and round to the nearest tenth: 4464/9000 x 100 = 49.6. So, $x = 49.6$. 