How many integers from 1 through 9999, inclusive, do not contain any of the digits 2, 3, 4 or 5?
We have 6 digits to choose from: 0, 1, 6, 7, 8, and 9.  We therefore have 6 choices for each of the digits in a 4-digit number, where we think of numbers with fewer than four digits as having leading 0s.  (For example, 0097 is 97.)  Since we have 6 choices for each of the four digits in the number, there are $6^4 = 1296$ ways to form the number.  However, we must exclude 0000 since this is not between 1 and 9999, inclusive,  so there are $1296-1 = \boxed{1295}$ numbers.
1295
Level 5
Counting & Probability
Problem: How many integers from 1 through 9999, inclusive, do not contain any of the digits 2, 3, 4 or 5?
Solution:<|START_OF_LATENT|><|PRIOR_PREFIX|>To determine how many integers from 1 to 9999 do not contain the digits 2, 3, 4, or 5, we first identify the allowable digits. The digits that can be used are 0, 1, 6, 7, 8, and 9. This gives us a total of 6 usable digits.

Next, we consider the number of digits in the integers we are counting. The integers can have 1 to 4 digits.

1. For 1-digit integers (1 to 9), the valid digits are 1, 6, 7, 8, and 9. This gives us 5 options.

2. For 2-digit integers (10 to 99), the first digit cannot be 0, so it can be 1, 6, 7, 8, or 9 (5 options). The second digit can be any of the 6 digits (0, 1, 6, 7, 8, 9). Thus, the total for 2-digit integers is:
   5 (choices for the first digit) * 6 (choices for the second digit) = 30.

3. For 3-digit integers (100 to 999), the first digit again cannot be 0, so it can be 1, 6, 7, 8, or 9 (5 options). The second and third digits can each be any of the 6 digits. Therefore, the total for 3-digit integers is:
   5 (choices for the first digit) * 6 (choices for the second digit) * 6 (choices for the third digit) = 180.

4. For 4-digit integers (1000 to 9999), the first digit cannot be 0, so it can be 1, 6, 7, 8, or 9 (5 options). The second, third, and fourth digits can each be any of the 6 digits. Thus, the total for 4-digit integers is:
   5 (choices for the first digit) * 6 (choices for the second digit) * 6 (choices for the third digit) * 6 (choices for the fourth digit) = 1080.

Now, we sum the counts from all digit lengths:
1-digit: 5
2-digit: 30
3-digit: 180
4-digit: 1080

Total = 5 + 30 + 180 + 1080 = 1295.

Thus, the final answer is 1295.<|END_OF_LATENT|> The final answer is $1295$.