We need to find out how many 6-digit sequences have a digit that shows up at least 5 times.
Step 1: Understand the problem A 6-digit sequence consists of 6 digits, each ranging from 0 to 9. Leading zeros are allowed, so a sequence like 000001 is valid. We are looking to count how many of these sequences contain a digit that appears at least 5 times. There are two cases to consider: 1. A digit appears exactly 5 times. 2. A digit appears exactly 6 times.
Step 2: Total number of 6-digit sequences Each position has 10 options (the digits 0 through 9), and with 6 positions, we find: Total sequences = 10^6 = 1,000,000.
Step 3: Case 1 – A digit appears exactly 6 times In this case, all digits in the sequence are the same, such as 111111, 000000, etc. There are 10 sequences like this—one for each digit from 0 to 9.
Step 4: Case 2 – A digit appears exactly 5 times Now let’s count how many of these sequences we have. First, choose the digit that appears 5 times. We have 10 choices (digits 0–9). Next, choose the position for the other digit (the one that appears once). There are 6 ways to choose which of the 6 positions is different. Finally, choose the digit that appears once. This can be any digit except the repeated one, giving us 9 choices. Total for this case = 10 × 6 × 9 = 540.
Step 5: Add both cases For Case 1 (all 6 digits the same): 10 For Case 2 (one digit appears 5 times): 540 So, the total number of sequences with a digit that appears at least 5 times equals 10 + 540 = 550. Final Answer: 550.
