How to Form 3-Digit Numbers Using Unique Digits: A Comprehensive Guide

When faced with the task of forming 3-digit numbers using a set of unique digits, a systematic approach is necessary. This guide will walk you through a step-by-step process to determine how many 3-digit numbers can be formed using each of the given digits 2, 3, 5, 7, and 9 exactly once. We will explore the mathematical concepts and provide detailed examples to enhance your understanding.

Understanding the Problem

The problem requires forming 3-digit numbers using the digits 2, 3, 5, 7, and 9, with each digit used only once. This is a classic permutation problem where order matters, and each digit can be used only once.

Step-by-Step Solution

Step 1: Selecting the Digits

The first step is to select 3 digits out of the 5 available. This can be done in a number of ways, but for clarity, we will consider the selection of any 3 digits out of the 5.

Formula: Combination

The formula for calculating the number of ways to choose 3 digits from 5 is given by the combination formula:

n  5 (total number of digits)
r  3 (number of digits to choose)
[_{n}C_{r}  frac{n!}{r!(n-r)!}  frac{5!}{3!(5-3)!}  frac{5 times 4}{2 times 1}  10]

Therefore, there are 10 ways to choose 3 digits from the 5 digits provided.

Step 2: Arranging the Selected Digits

Once the 3 digits are chosen, the next step is to arrange them in different orders. The number of ways to arrange 3 digits is given by the factorial of 3 (3!):

3!  3 times 2 times 1  6]

Therefore, each set of 3 chosen digits can be arranged in 6 different ways.

Step 3: Calculating the Total Number of 3-Digit Numbers

The total number of 3-digit numbers can be calculated by multiplying the number of ways to choose the digits by the number of arrangements of those digits:

Total 3-digit numbers  [_{5}C_{3} times 3!  10 times 6  60]

Thus, the total number of 3-digit numbers that can be formed using the digits 2, 3, 5, 7, and 9, with each digit used only once, is 60.

Example: Forming 3-Digit Numbers

Let’s list the 3-digit numbers formed by the permutation of the digits 2, 3, 5, 7, and 9:

235, 253, 325, 352, 523, 532 237, 273, 327, 372, 723, 732 239, 293, 329, 392, 923, 932 257, 275, 527, 572, 725, 752 259, 295, 529, 592, 925, 952 279, 297, 729, 792, 927, 972 357, 375, 537, 573, 735, 753 359, 395, 539, 593, 935, 953 379, 397, 739, 793, 937, 973 579, 597, 759, 795, 957, 975

This exhaustive list confirms that there are indeed 60 unique 3-digit numbers that can be formed.

Generalization to Other Cases

The method described can be generalized to any set of digits. For example:

General Formula: Permutation

If you have n digits and want to form r digit numbers, the formula for the total number of permutations is:

pnr  [frac{n!}{(n-r)!}]

This formula is used to calculate the number of distinct permutations of a subset of r items from a set of n items.

Conclusion

In this guide, we have demonstrated how to determine the number of 3-digit numbers that can be formed using unique digits from a set. By understanding and applying the combination and permutation formulas, you can easily solve similar problems involving permutations and combinations.

Combination: The number of ways to choose r items from a set of n items without regard to order. Permutation: The number of ways to arrange r items from a set of n items, considering the order of selection.

By following these steps, you can systematically solve a wide range of permutation problems and enhance your problem-solving skills in mathematics and beyond.