Final answer:
To find the number of unique 8-digit strings with specific conditions, we calculate combinations by arranging the required digits and then multiplying by the number of options for the free spot. This results in a total of 3360 unique strings.
Step-by-step explanation:
We need to find the number of 8-digit strings that contain the digit 5 exactly once, the digit 2 exactly twice, and the digit 7 exactly thrice, with more 6s than 2s. This means that we need at least three 6s in our number because we have two 2s. Given these conditions, we have one free spot in the 8-digit string which can be filled by digits 0, 1, 3, 4, 8, or 9.
First, we calculate the number of ways to position our fixed digits (5, two 2s, three 7s, and three 6s). There are 8! ways to arrange eight items, but since we have repeats, we divide by the factorial of the number of repeats for each digit: 8!/(1!*2!*3!*3!).
Next, we consider the one free spot available. Since it can be filled by any of six digits, we multiply the previously calculated number by 6. Therefore, the total number of strings is 8!/(1!*2!*3!*3!) * 6.
Calculations: 8! = 40320, 1! = 1, 2! = 2, 3! = 6. Therefore, 40320/(1*2*6*6) equals 560. Multiplying 560 by 6 for the free spot gives us 3360 unique 8-digit strings.