Question

How many odd numbers between 1000 and 10,000 have no digits repeated?

Short answer

There are 2520 odd numbers between 1000 and 10,000 with no repeating digits.

Step-by-step solution

Determine the Number of Digits

We are looking for odd numbers between 1000 and 10,000, which means the numbers must have 4 digits, with the thousandth digit being non-zero.

Restriction on the Last Digit (Odd)

The last digit of the number must be odd, which can be 1, 3, 5, 7, or 9. So, we have 5 options for the last digit.

Choose a Number for the Thousand's Place

For the first digit, which can be any digit from 1 to 9 (since it cannot be zero), we pick one of the remaining 9 digits such that it is different from the selected unit digit.

Choose a Number for the Hundred's Place

For the hundred's place, select any remaining digit. We now have 8 options since we can't repeat the thousand's and unit's place digits.

Choose a Number for the Ten's Place

Similar to previous steps, choose a digit for the tens place from the 7 remaining options, as it should differ from the previously chosen digits.

Calculate Total Combinations

We multiply the choices for each place: 5 choices for the units digit, 9 for the thousand’s place, 8 for the hundred’s, and 7 for the ten’s to get the total combinations as follows: \[5 \times 9 \times 8 \times 7\].

Compute the Result

Multiply these numbers: \[5 \times 9 = 45\], \[45 \times 8 = 360\], and \[360 \times 7 = 2520\]. Therefore, the total number of such numbers is 2520.

Key concepts

Odd Numbers

Odd numbers are integers that are not divisible by 2. In simpler terms, odd numbers have a remainder of 1 when divided by 2. For instance, numbers like 1, 3, 5, 7, and 9 are odd. They play a crucial role when you are tasked with finding specific number patterns, such as numbers between certain values.
In the provided exercise, we focus on four-digit numbers, meaning numbers ranging from 1000 to 9999. The task is to find those numbers ending in an odd digit, which categorizes them distinctly from even numbers. Remember, an even number will end in 0, 2, 4, 6, or 8, contrary to what our exercise requires.

Digit Restriction

Digit restriction means limiting which digits can be used in certain places for a particular combination problem. In this exercise, the problem is limiting repeated use of the same digit in forming a 4-digit odd number.
Starting from the last digit, which is crucial for ensuring the number is odd, it can only be 1, 3, 5, 7, or 9. After picking a digit for the unit's place, we must carefully select other digits for the remaining positions (thousand's, hundred's, and ten's places) while ensuring each digit is unique. This constraint significantly influences how we approach our selections and prevents repetition across the number.

Combinatorial Counting

Combinatorial counting is a mathematical strategy to determine the number of ways to choose or arrange items under certain conditions. In exercises like this one, we use combinatorial counting to find possible four-digit combinations that meet the given criteria.
The challenge is to count the numbers that are odd and do not have repeating digits in each position. This counting is achieved by multiplying available options for each digit position. The crucial tip here is knowing how to move from one decision to another, gradually reducing choices as we fill each digit position while meeting the restrictions.

Permutation without Repetition

Permutation without repetition concerns arranging distinct items where each item is unique and does not reappear in any arrangement. In this numeral context, each digit used should only appear once in the number.
In our solution, the permutation starts by selecting an odd digit for the unit's place, then proceeds to choose remaining digits in sequence for the thousand's, hundred's, and ten's places. The absence of repetition leads to decrementing options at each step: first picking from all odd numbers, then reducing available choices for subsequent digits, ensuring a unique combination throughout.