How many numbers between 100 and 1000 have the digit 6 exactly once?

There are 225 numbers between 100 and 1000 that contain the digit 6 exactly once.

Here's a breakdown of why:

• Understanding the Problem: We need to count numbers within the range of 100 to 999 (inclusive) where the digit 6 appears only once.

• Analyzing Possibilities: Since we're dealing with three-digit numbers, we have three places where the digit 6 could appear: hundreds, tens, or ones place. We'll consider these separately.

  1. 6 in the Hundreds Place (6XY):

     • The hundreds place is fixed as 6.
     • The tens place (X) cannot be 6; it can be any of the other 9 digits (0-9 excluding 6).
     • The ones place (Y) also cannot be 6; it can be any of the other 9 digits (0-9 excluding 6).
     • This gives us 9 * 9 = 81 possibilities.

  2. 6 in the Tens Place (X6Y):

     • The hundreds place (X) cannot be 0 or 6; it can be any of the 8 digits (1-9 excluding 6).
     • The tens place is fixed as 6.
     • The ones place (Y) cannot be 6; it can be any of the other 9 digits (0-9 excluding 6).
     • This gives us 8 * 9 = 72 possibilities.

  3. 6 in the Ones Place (XY6):

     • The hundreds place (X) cannot be 0 or 6; it can be any of the 8 digits (1-9 excluding 6).
     • The tens place (Y) cannot be 6; it can be any of the other 9 digits (0-9 excluding 6).
     • The ones place is fixed as 6.
     • This gives us 8 * 9 = 72 possibilities.

• Total Count: We add the possibilities from each case: 81 + 72 + 72 = 225

Therefore, there are 225 numbers between 100 and 1000 that have the digit 6 exactly once. This matches the information provided in the reference.