Question

Сколько целых чисел от 0 до 99999 не содержат двух цифр 2 подряд в своей десятичной записи?

Answer 1

Ты что решаешь олимпиаду высшей пробы ?)

Answer 2

Number of integers from 0 to 99999 without two consecutive 2s

To determine the number of integers from 0 to 99999 that do not contain two consecutive 2s in their decimal representation, we can use a counting approach.

Let's break down the problem into smaller cases:

1. Numbers with 1 digit: There are 10 single-digit numbers (0 to 9), and none of them contain two consecutive 2s.

2. Numbers with 2 digits: There are 90 two-digit numbers (10 to 99). We need to count how many of them do not have two consecutive 2s. Since the first digit can be any digit from 1 to 9, there are 9 possibilities. For the second digit, it can be any digit from 0 to 9 except 2. Therefore, there are 9 * 9 = 81 two-digit numbers without two consecutive 2s.

3. Numbers with 3 digits: There are 900 three-digit numbers (100 to 999). Again, we need to count how many of them do not have two consecutive 2s. The first digit can be any digit from 1 to 9, giving us 9 possibilities. For the second and third digits, they can be any digit from 0 to 9 except 2. Therefore, there are 9 * 9 * 9 = 729 three-digit numbers without two consecutive 2s.

4. Numbers with 4 digits: There are 9000 four-digit numbers (1000 to 9999). Using the same logic as above, there are 9 * 9 * 9 * 9 = 6561 four-digit numbers without two consecutive 2s.

5. Numbers with 5 digits: There are 90000 five-digit numbers (10000 to 99999). Again, using the same logic, there are 9 * 9 * 9 * 9 * 9 = 59049 five-digit numbers without two consecutive 2s.

To find the total number of integers from 0 to 99999 without two consecutive 2s, we sum up the counts from each case:

10 + 81 + 729 + 6561 + 59049 = 66430

Therefore, there are 66430 integers from 0 to 99999 that do not contain two consecutive 2s in their decimal representation.