Introduction to Forming Five-Digit Numbers

To understand how many five-digit numbers can be formed using the digits 1, 2, 3, 4, 5, 6, and 7 without repeating any digits, we need to use the fundamental concept of permutations. Permutations are about arranging a set of items where the order matters and no items are repeated.

Step-by-Step Explanation:

Let's break down the process of forming these numbers:

Choosing the First Digit:

For the first digit, we have 7 options (1, 2, 3, 4, 5, 6, and 7).

Choosing the Second Digit:

Once the first digit is chosen, we have 6 remaining options for the second digit.

Choosing the Third Digit:

After selecting the first two digits, 5 options remain for the third digit.

Choosing the Fourth Digit:

After choosing the first three digits, 4 options are available for the fourth digit.

Choosing the Fifth Digit:

Finally, after selecting the first four digits, we have 3 options left for the fifth digit.

The total number of five-digit combinations is calculated as follows:

7 x 6 x 5 x 4 x 3 2520

Let's perform the calculation step-by-step:

7 x 6 42 42 x 5 210 210 x 4 840 840 x 3 2520

Therefore, the total number of five-digit numbers that can be formed is 2520. This calculation ensures that each digit is used only once in each number, making it a perfect application of permutations.

Other Methods of Solution

Another way to look at this problem involves viewing the formation of the five-digit numbers as a sequence of choices:

The first place can be filled in 7 ways. The second place can be filled in 6 ways. The third place can be filled in 5 ways. The fourth place can be filled in 4 ways. The fifth place can be filled in 3 ways.

The total number of sequences is 7 x 6 x 5 x 4 x 3 2520.

Application to Smaller Sets

For a smaller set of numbers, such as 1, 2, 4, 6, and 8, we can apply the same principles:

Smallest Five-Digit Number: The smallest number is formed by arranging the digits in ascending order, which is 12468.

Largest Five-Digit Number: The largest number is formed by arranging the digits in descending order, which is 86421.

Using the same method, the total number of combinations for these digits is:

5 x 4 x 3 x 2 x 1 120 ways

The calculation for the number of combinations for any set of 5 distinct digits can be represented by the formula for permutations: Pn,r n! / (n-r)! 5! / (5-5)! 5! 120.

Conclusion

The concept of permutations is crucial in solving such problems, where the order of digits matters and no digit is to be repeated. Understanding and applying this concept helps in solving similar problems efficiently.