Exploring the Number of Three-Digit Combinations with the Digits 1, 7, and 3

Counting the number of three-digit combinations possible using the digits 1, 7, and 3 can seem like a straightforward problem. However, the subtleties involved in whether digit repetition is allowed or not can greatly affect the solution. Let's delve into the details and explore various scenarios.

When Digit Repetition is Allowed

In scenarios where the digits 1, 7, and 3 can be repeated, we need to consider the three positions in a three-digit number: the hundreds, tens, and units. Each of these positions can be filled by any of the three digits.

We can calculate the total combinations using the following formula:

( text{Total combinations} 3 times 3 times 3 27 )

Let's list some of the possible combinations to illustrate:

111 113 117 131 133 137 171 173 177 333 331 337 313 311 317 373 371 377 777 771 773 717 711 713 737 731 733

When Digit Repetition is Not Allowed

When repetitions are not allowed, each digit can only appear once in each number. In this case, we need to consider permutations of the three digits taken three at a time. The formula for permutations without repetition is given by:

( P(3, 3) 3! 3 times 2 times 1 6 )

The possible permutations are:

137 173 317 371 713 731

Using Mathematical Symbols for Varied Representations

Mathematics allows us to explore numbers in various ways. Here are some examples:

(7^{1^3} 343) (7^{3^1} 343)

Surprisingly, these two expressions yield the same value, even though they are written differently. More complex representations can include summations and factorials:

(sum_{i1}^{7}3!)

These equations provide alternative ways to express and understand the number 343.

Conclusion

The number of three-digit combinations using the digits 1, 7, and 3 can vary greatly depending on whether repetitions are allowed. When repetition is allowed, there are 27 combinations, while when it is not, there are only 6. Mathematics is rich with such nuances and possibilities, offering us endless ways to explore and represent numbers.