Question

Find the number of all 20 digit integers in which no two consecutive digits are the same

Answer 1

For the first digit of the required 20-digit number, we can have 9 choices for the number, those are the digits 1 to 9. In the second digit of the number, we will only have 8 choices. That is because, we don't want to write a number in the second digit that is similar to the first digit and the third digit.

Similarly, in the third digit's place, we will only have 8 choices because we do not want the digit to be similar to the second digit and the fourth digit. This will go on up until the 19th digit. In the 20th digit, we will also have 9 choices (from 0-9) but we do not want the digit to be similar to the 19th digit. 

From the explanation given above, we have the number of ways,

    n = 9 x 8^(19 - 1) x 9  = 1.459 x 10^18

ANSWER: 1.459 x 10^18