How many four-digit odd numbers less than 6000 can be formed using the digits 2, 4, 6, 7, 8, and 9?

Verified step by step guidance

1. Identify the constraints for the four-digit number: it must be less than 6000, so the first digit can be 2, 4, or any digit less than 6 from the given set. Since the digits given are 2, 4, 6, 7, 8, and 9, the first digit can only be 2 or 4 to satisfy the 'less than 6000' condition.
2. Determine the possible digits for the last digit since the number must be odd. From the given digits, the odd digits are 7 and 9, so the last digit can be either 7 or 9.
3. For the second and third digits, there are no restrictions other than using the given digits. So, each of these positions can be any of the 6 digits: 2, 4, 6, 7, 8, or 9.
4. Calculate the total number of such numbers by multiplying the number of choices for each digit: number of choices for the first digit × number of choices for the second digit × number of choices for the third digit × number of choices for the last digit.
5. Express the total count as: $2\times 6\times 6\times 2$, where 2 is for the first digit options (2 or 4), 6 for the second digit, 6 for the third digit, and 2 for the last digit (7 or 9).

Key Concepts

Counting Principles and Permutations

Counting principles help determine the number of ways to arrange or select items under given constraints. In this problem, permutations with repetition are considered since digits can be reused unless stated otherwise. Understanding how to count valid combinations based on positional restrictions is essential.

Place Value and Number Formation

Place value determines the value of each digit depending on its position in a number. For four-digit numbers, the first digit represents thousands, which affects the range (less than 6000 means the first digit must be 2, 4, or another digit less than 6). Recognizing place value constraints guides valid digit choices.

Properties of Odd Numbers

An odd number ends with an odd digit. Identifying which digits are odd (in this case, 7 and 9) is crucial for determining the possible last digits of the number. This property restricts the choices for the units place and influences the total count of valid numbers.