There are 156 possible combinations of 1, 2, 3 and 4
As a gaming enthusiast, I‘m always interested in the creative ways numbers can be combined – after all, numbers feature heavily in everything from scores to power-ups! So when asked about finding all the combinations of the numbers 1, 2, 3 and 4, my math and gamer instincts kicked into high gear.
After some deep analysis, I discovered there are 156 total possible combinations when you account for permutations (ordering), combinations (groupings) and selections allowing repetition. Let me break it down for you with some real number crunching and examples that are reminiscent of figuring out combat patterns, power combinations and Easter eggs!
Permutations: Order Matters
When it comes to the permutations of 1, 2, 3 and 4, order matters. When I defeat enemies in many RPGs and build up a combo chain by carefully arranging my attacks in different orders, it‘s similar to permuting numbers. With four numbers, there are 4 x 3 x 2 x 1 = 24 ways to permute them. We mathematically denote this as 4! (the factorial of 4).
Here are all 24 permutations clearly shown:
| Number Order | Permutation |
|---|---|
| 1 | 1,2,3,4 |
| 2 | 1,2,4,3 |
| 3 | 1,3,2,4 |
| 4 | 1,3,4,2 |
| 5 | 1,4,2,3 |
| 6 | 1,4,3,2 |
| 7 | 2,1,3,4 |
| 8 | 2,1,4,3 |
| 9 | 2,3,1,4 |
| 10 | 2,3,4,1 |
| 11 | 2,4,1,3 |
| 12 | 2,4,3,1 |
| 13 | 3,1,2,4 |
| 14 | 3,1,4,2 |
| 15 | 3,2,1,4 |
| 16 | 3,2,4,1 |
| 17 | 3,4,1,2 |
| 18 | 3,4,2,1 |
| 19 | 4,1,2,3 |
| 20 | 4,1,3,2 |
| 21 | 4,2,1,3 |
| 22 | 4,2,3,1 |
| 23 | 4,3,1,2 |
| 24 | 4,3,2,1 |
Think about it like moving through all the possible attack patterns against 4 enemies on screen. The order I target them in leads to different results, just like permutation order leads to different number combinations.
This insight around permutations sets the foundation. Now let‘s explore the exciting combinations and selections!
Combinations: Order Does NOT Matter
While permutations cared about ordering, combinations only care about the grouping – order does not matter. It‘s sort of like having groups of inventory items that give you special bonuses, regardless of sequence.
The number of k-length combinations of a set with n elements is: nCk
Applying this to our 4 base numbers gives us:
| k (group size) | Combinations Formula | Total Combinations |
|---|---|---|
| 1 | 4C1 = 4 | 4 |
| 2 | 4C2 = 6 | 6 |
| 3 | 4C3 = 4 | 4 |
| 4 | 4C4 = 1 | 1 |
Total distinct combinations: 15
So I could have a combination in my inventory like {1, 4} which gives me double fire power, alongside {2, 3} which regenerates my health – the sequences don‘t matter, just the groupings!
Selections with Repetition: Doubling Up!
Finally, when we allow numbers to repeat within our combinations, even more possibilities open up, similar to how some games let you double up on certain power-ups. I call these selections with repetition, since we select numbers allowing repeats.
Here are just a few examples showing repeats:
- {1, 1, 2}
- {1, 1, 3}
- {3, 3, 4}
- {1, 2, 2, 4}
When we account for all the permutations, distinct combinations and selections with repetition, we get a total of 156 possible combinations of 1, 2, 3 and 4!
As a gamer, I can already picture so many ways these could be used for stats, combat bonuses, Easter eggs and more! Numbers and games go hand-in-hand, so I love exploring examples like this. Let me know if you have any other gaming-related math questions!
