Understanding the Number of 5-Digit Combinations from 1 to 15

When considering the number of 5-digit combinations that can be made using the numbers 1 through 15, it is essential to distinguish between the conditions of repetition and order. This article will delve into each of the four primary cases: with repetition and order matters, with repetition and order does not matter, no repetition and order matters, and no repetition and order does not matter.

Case 1: Repetition Allowed, Order Matters

In the first scenario, we allow the repetition of digits and consider each digit's position in the 5-digit sequence as significant. Since there are 15 options for each of the 5 digits, we use the formula for permutations:

155 759,375

This means that there are 759,375 unique 5-digit combinations when repetition is allowed and order matters. Each of the 5 positions can independently take any of the 15 values, leading to a huge number of possibilities.

Case 2: Repetition Allowed, Order Does Not Matter

In this scenario, repetition of digits is still allowed, but the order in which the digits appear does not matter. To solve this, we use the binomial coefficient (combinations) formula:

( binom{15 5-1}{5} binom{19}{5} frac{19!}{5!(19-5)!} frac{19 times 18 times 17 times 16 times 15}{5 times 4 times 3 times 2 times 1} 11,628 )

Here, the formula ( binom{19}{5} ) represents the number of ways to choose 5 items from 19 without regard to order. This results in 11,628 distinct combinations.

Case 3: No Repetition, Order Matters

When no repetition of digits is allowed and the order matters, we calculate the number of permutations:

( P(15, 5) frac{15!}{(15-5)!} frac{15!}{10!} 15 times 14 times 13 times 12 times 11 360,360 )

This formula represents the number of ways to choose 5 digits from 15 and arrange them in a specific order. The result is 360,360 unique 5-digit combinations.

Case 4: No Repetition, Order Does Not Matter

Finally, when no repetition of digits is allowed and the order does not matter, we calculate the number of combinations:

( binom{15}{5} frac{15 times 14 times 13 times 12 times 11}{5 times 4 times 3 times 2 times 1} 3,003 )

This formula counts the number of ways to choose 5 items from 15 without regard to order. The result is 3,003 unique combinations.

Summary

- With repetition allowed and order matters: 759,375

- With repetition allowed and order does not matter: 11,628

- No repetition and order matters: 360,360

- No repetition and order does not matter: 3,003

Note: Each of the conditions significantly alters the number of possible combinations, highlighting the importance of considering both repetition and order in such calculations.