Question

1)Автобусные билеты имеют номера от 000 001 до 999 999. Номер считается счастливым, если три первые его цифры нечётны и различны, вторые три цифры четны, причем 7 и 8 не стоят рядом. Сколько существует различных счастливых номеров? 1) 6588; 2) 7748; 3) 7200; 4) 5610.

Answer 1

To find the number of different lucky numbers that satisfy the given conditions, we need to break down the conditions and calculate the possibilities for each part.

1. The first three digits must be odd and different from each other. - There are 5 odd digits (1, 3, 5, 7, 9) to choose from for the first digit. - After choosing the first digit, there are 4 odd digits remaining to choose from for the second digit. - After choosing the first two digits, there are 3 odd digits remaining to choose from for the third digit. - Therefore, the number of possibilities for the first three digits is 5 * 4 * 3 = 60.

2. The next three digits must be even. - There are 5 even digits (0, 2, 4, 6, 8) to choose from for each of the three digits. - Therefore, the number of possibilities for the next three digits is 5 * 5 * 5 = 125.

3. The digits 7 and 8 cannot be adjacent. - There are 5 possible positions for the digit 7 (before the first even digit, between the first and second even digits, between the second and third even digits, after the third even digit, or not present). - After placing the digit 7, there are 4 possible positions for the digit 8 (before the first even digit, between the first and second even digits, between the second and third even digits, or after the third even digit). - Therefore, the number of possibilities for the positions of the digits 7 and 8 is 5 * 4 = 20.

To find the total number of different lucky numbers, we multiply the number of possibilities for each part together: 60 * 125 * 20 = 150,000.

Therefore, there are 150,000 different lucky numbers that satisfy the given conditions.

Answer: 150,000.