How Many 3-Digit Numbers Can Be Formed Using the Digits 1 to 7 without Repetition?

When forming 3-digit numbers using the digits 1 through 7 without repeating any digit, the number of possible combinations can be calculated systematically. This article will explore the methods and provide a step-by-step breakdown of the solution.

Logical Order and Choices for Digits

Firstly, let's consider the logical order of forming these 3-digit numbers. A 3-digit number is represented as _ _ _. For each of these positions, you have specific choices of digits from the set {1, 2, 3, 4, 5, 6, 7}.

Hundreds Place

The hundreds place can be filled by any one of the 7 digits. Thus, we have 7 choices for the leftmost digit.

Tens Place

For the tens place, we have 6 choices left since one digit has already been used for the hundreds place. Hence, the number decreases by 1.

Units Place

For the units place, we have 5 choices remaining as two digits have already been used. The count decreases by another 1.

Thus, the total number of 3-digit numbers that can be formed is:

7 (choices for hundreds) × 6 (choices for tens) × 5 (choices for units) 210.

The Theoretical Approach: Permutations

Another way to approach this problem is by considering it as a permutation problem. For a 3-digit number, we have 3 positions which can be filled by 7 unique digits without repetition.

The formula for permutations is given by nPr n! / (n-r)!, where n is the total number of items, and r is the number of items to be selected. Here, n 7 (the digits 1 to 7) and r 3 (the 3 digits in the number).

P3,7 7! / (7-3)! 7! / 4! (7 × 6 × 5 × 4 × 3 × 2 × 1) / (4 × 3 × 2 × 1) 7 × 6 × 5 210.

Mathematical Series and Numbering Analysis

To provide a more detailed understanding, let's explore the series of numbers formed:

Sequence Start:

The sequence starts with 123 and continues up to 765. Each digit from 1 to 7 can be placed in the hundreds, tens, and units places, ensuring no digit is repeated.

Total Count:

By the principle of multiplication, the total count of 3-digit numbers is calculated as: