While writing all the numbers from 700 to 1000, how many numbers occur in which the digit at hundred’s place is greater than the digit at ten’s…

Correct Option (3)

The problem requires identifying numbers between 700 and 1000 (i.e., from 700 to 999) that satisfy two conditions:

• The digit at the hundreds place (H) is greater than the digit at the tens place (T). (H > T)
• The digit at the tens place (T) is greater than the digit at the units place (U). (T > U)

Given the range, the hundreds digit (H) can only be 7, 8, or 9. We analyze each case:

Case 1: Hundreds digit (H) = 7

• Since H > T, T must be less than 7 (i.e., 0, 1, 2, 3, 4, 5, 6).
• Since T > U, U must be less than T. For U to be a valid digit (0-9), T must be at least 1.
• If T = 1, U can be 0 (1 number).
• If T = 2, U can be 0, 1 (2 numbers).
• If T = 3, U can be 0, 1, 2 (3 numbers).
• If T = 4, U can be 0, 1, 2, 3 (4 numbers).
• If T = 5, U can be 0, 1, 2, 3, 4 (5 numbers).
• If T = 6, U can be 0, 1, 2, 3, 4, 5 (6 numbers).
• Total numbers for H=7: 1 + 2 + 3 + 4 + 5 + 6 = 21.

Case 2: Hundreds digit (H) = 8

• Since H > T, T must be less than 8 (i.e., 0, 1, 2, 3, 4, 5, 6, 7).
• Again, T must be at least 1 for U to exist.
• The number of unit digits (U) for each tens digit (T) follows the same pattern:
  • If T = 1, U can be 0 (1 number).
  • ...
  • If T = 7, U can be 0, 1, 2, 3, 4, 5, 6 (7 numbers).
• Total numbers for H=8: 1 + 2 + 3 + 4 + 5 + 6 + 7 = 28.

Case 3: Hundreds digit (H) = 9

• Since H > T, T must be less than 9 (i.e., 0, 1, 2, 3, 4, 5, 6, 7, 8).
• T must be at least 1.
• The number of unit digits (U) for each tens digit (T) follows the same pattern:
  • If T = 1, U can be 0 (1 number).
  • ...
  • If T = 8, U can be 0, 1, 2, 3, 4, 5, 6, 7 (8 numbers).
• Total numbers for H=9: 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 = 36.

The total number of such numbers is the sum of numbers from all cases:

Total = 21 (for H=7) + 28 (for H=8) + 36 (for H=9) = 85.

Incorrect Options:

Options 1 (61), 2 (64), and 4 (91) are incorrect. These values do not correspond to the systematic enumeration of numbers satisfying the given conditions. The calculation method ensures that all valid combinations are counted precisely, leading to the unique correct total of 85.