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How many six digit numbers do not have 3 consecutive digits the same

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Final answer:

To find the number of six-digit numbers that do not have 3 consecutive digits the same, use the concept of permutations. Follow the steps of choosing each digit and excluding repetitions. There are 4,608,000 such numbers.

Step-by-step explanation:

In order to find the number of six-digit numbers that do not have 3 consecutive digits the same, we can use the concept of permutations. We have 10 choices for each digit: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. However, if we choose a digit, we cannot choose the same digit for the next two places. Let's break it down step-by-step:

  1. Choose the first digit: We have 10 choices.
  2. Choose the second digit: We also have 10 choices, but we must exclude the digit we chose for the first place. So we have 9 choices left.
  3. Choose the third digit: We have 10 choices again, but we must exclude the digit we chose for the second place as well as any digit repeated twice. So we have 8 choices left.
  4. Choose the remaining three digits: We apply the same logic as the previous steps. We have 10 choices for each digit, but we must exclude the digits repeated twice or thrice before. So we have 8 choices for the fourth digit, 8 choices for the fifth digit, and 8 choices for the sixth digit.

To find the total number of possibilities, we multiply the number of choices at each step together: 10 choices for the first digit × 9 choices for the second digit × 8 choices for the third digit × 8 choices for the fourth digit × 8 choices for the fifth digit × 8 choices for the sixth digit. This gives us a total of 4,608,000 six-digit numbers that do not have 3 consecutive digits the same.

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