Understanding Permutations: How Many Combinations Are There in 12345?

How many possible permutations does the sequence 12345 have?

The answer is 120, not 60. This is because we are dealing with permutations, not combinations. In mathematical terms, permutations are different arrangements of the same set of items. Let's explore how we arrived at this number.

The Mathematical Explanation

Imagine forming a generic 5-digit number using the digits 1, 2, 3, 4, and 5 once and exactly once. The first digit has 5 options—it could be 1, 2, 3, 4, or 5. After the first digit has been chosen, we are left with 4 options for the second digit, as we can't reuse the first digit. Following this pattern:

1 option for each of the remaining digits gives us: 4, 3, 2, and 1 respectively.

Thus, we multiply the number of options at each step: 5 × 4 × 3 × 2 × 1 120.

This calculation can also be represented with a tree diagram. Each branch represents a choice for a digit, and the number of unique pathways at the end of the tree corresponds to the number of permutations. For a more in-depth understanding, I recommend looking up tree diagrams in mathematical resources.

Visualizing the Permutations

Let's list some of the permutations to visualize the possibilities:

12345 23451 34512 45123 51234 54321 43215 32154 21543 15432 15243 51423 34215 45321

There are indeed a lot of permutations, as identified by Mal in his comment.

Breaking Down the Permutations

Assuming you fix the first two digits as 12, we can see that there are 6 permutations of the remaining three digits (3, 4, 5). The permutations with 1 and 2 fixed in the first two positions are:

12345 12354 12435 12453 12534 12543

By repeating this process for every possible pair of fixed digits (12, 13, 14, 15, 23, etc.), we can calculate the total number of permutations:

Fixing 12: 6 permutations Fixing 13: 6 permutations Fixing 14: 6 permutations Fixing 15: 6 permutations Fixing 23: 6 permutations Fixing 24: 6 permutations Fixing 25: 6 permutations Fixing 34: 6 permutations Fixing 35: 6 permutations Fixing 45: 6 permutations

Each of these sets of permutations gives us a total of 6 permutations, and as there are 5 possible pairs of fixed digits, the total permutations are 6 × 5 30. By extending this to fixing any of the first two digits, we get 5 × 6 30 permutations for each of the first digit.

Since we need to consider the first digit, we multiply by the number of first digit options (5) giving us: 24 × 5 120 total permutations.

Conclusion

In summary, the number of permutations of the sequence 12345 is 120. This is a rich example of permutations and can be helpful in understanding more complex mathematical concepts. For those looking to delve deeper into this topic, further exploration of tree diagrams and permutations is recommended.