The sum of all 8-digit numbers formed using the digits 1, 2, 3, 4, 5, 6, 7 exactly once (with one digit repeated twice and the others once) such that the units digit is less than the tens digit is 3,024,000,786,240.
This result is obtained by considering the possible pairs for the tens and units digits (with tens > units), the choice of the repeated digit, and the sum of the numbers formed by the remaining six digits. The calculation accounts for all cases and symmetries, yielding the final sum.
3024000786240}
The sum of all 8-digit numbers formed using the digits 1, 2, 3, 4, 5, 6, 7 exactly once (with one digit repeated twice and the others once) such that the units digit is less than the tens digit is 3,024,000,786,240.
This is calculated by summing over all valid configurations, considering the choices for the tens and units digits (with tens > units), the repeated digit, and the arrangements of the remaining six digits. The total number of such numbers is 68,040, and their sum is computed as detailed.
3024000786240}
The sum of all 8-digit numbers formed using the digits 1, 2, 3, 4, 5, 6, 7 exactly once (with one digit repeated twice and the others once) such that the units digit is less than the tens digit is 3,024,000,786,240.
This result is obtained by:
— Considering all ordered pairs (A, B) for the tens and units digits with A > B, where A and B are digits from 1 to 7.
— For each pair, accounting for the three cases of the repeated digit: neither A nor B, equal to A, or equal to B.
— Calculating the sum of the numbers formed by the remaining six digits in the higher places for each case, using the sum of place values and the sum of digits in each position.
— Summing over all pairs and cases, leading to the total sum.
The calculation verifies that the total number of such numbers is 68,040, and the sum is consistent with the symmetry and constraints.
3024000786240}