How many three digit numbers can be formed using the digits 1,2,3,4,5,6,7 and 8 without repeating the digits and such that the tens digit is greater than the hundreds digit and less than the units digit?A48B56C64D72

To solve this problem, we need to consider the conditions given:

1. The number is a three-digit number.
2. The digits 1,2,3,4,5,6,7,8 are used without repetition.
3. The tens digit is greater than the hundreds digit and less than the units digit.

Let's break it down:

Step 1: Since the tens digit has to be greater than the hundreds digit, we have 7 choices for the hundreds digit (1,2,3,4,5,6,7) because if we choose 8, there will be no greater number for the tens digit.

Step 2: After choosing the hundreds digit, we have 7 remaining choices for the tens digit. But since the tens digit has to be less than the units digit, we actually have 6 choices for the tens digit.

Step 3: After choosing the hundreds and tens digits, we have 6 remaining choices for the units digit.

So, the total number of three-digit numbers that can be formed is 7 (choices for hundreds digit) * 6 (choices for tens digit) * 6 (choices for units digit) = 252.

However, we have over-counted. For each choice of hundreds and units digits, we have counted every possible tens digit. But the problem states that the tens digit must be less than the units digit and greater than the hundreds digit. So for each choice of hundreds and units digits, there is exactly one choice of tens digit. Therefore, we have to divide our previous answer by the number of choices for the tens digit, which is 6.

So, the correct answer is 252 / 6 = 42.

Therefore, none of the options A, B, C, D is correct. The correct answer should be 42.