Find the number of integers from $0$ to $999999$ that have no two equal neighboring digits in their decimal representation.

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Solution

Find the total number digits from $0$ to $999999$ .
As we know that an integer is a whole number from a set of negative, non-negative, positive and $0$ numbers.
Case-1 From 0 to 9
the total number of digits from $0$ to $9$ having no neighbor common is equal to $10$
Case-2 The total number of digits from $10$ to $99$ having no neighbor common,
$= 9 \times 9$
$= 81$
Case-3 From $100$ to $999$
$= 9 \times 9 \times 9 = 9^{3}$
Case-4 From $1000$ to $9999$
$= 9 \times 9 \times 9 \times 9 = 9^{4}$
Case-5 From $10000$ to $99999$
$equals 9 cross times 9 cross times 9 cross times 9 cross times 9 equals 9 to the power of 5$
Case-6 From $100000$ to $999999$
$= 9 \times 9 \times 9 \times 9 \times 9 \times 9 = 9^{6}$
Total number of digits $= 10 + 9^{2} + 9^{3} + 9^{4} + 9^{5} + 9^{6}$
Hence, the total number of digits are $10 + 9^{2} + 9^{3} + 9^{4} + 9^{5} + 9^{5}$ .

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Find the number of integers from 0 to 999999 that have no two equal neighboring digits in their decimal representation.

Find the number of integers from $0$ to $999999$ that have no two equal neighboring digits in their decimal representation.