How Many Combinations of 1-6 Are There? 720!

As an avid gamer and creator, I love exploring the math behind puzzles and combinations in the games I play and design. One recent question that caught my attention: how many unique combinations can you make using the digits 1, 2, 3, 4, 5 and 6 if repetition is allowed? After digging into the underlying combinatorics concepts, I‘ve determined there are 720 total combinations – let‘s walk through the methods to derive this solution from a game design perspective!

Combinatorics Crash Course

Before solving specific combination problems, understanding some key fundamentals opens up loads of possibilities for crafting puzzles.

Factorials

The factorial (!) of a number N is defined as:

N! = N (N-1) (N-2) … 1

So for example:

5! = 5 4 3 2 1 = 120

Factorials show up often in counting problems. As we‘ll see, they enumerate permutations and combinations.

Permutations vs Combinations

Permutations count arrangements, where order matters.

Combinations count groups where internal order doesn‘t matter.

Key differences:

• Permutations use factorials
• Combinations use binomial coefficients

Consider selecting 3 cards out of 5 total cards.

The number of…

• Permutations: 5! / (5-3)! = 60 (order matters)
• Combinations: C(5,3) = 10 (only which 3 cards matter)

With some simple building blocks, we can count lots of possibilities!

And this leads us to…

Solving for 1-6 Combinations

Using the digits 1, 2, 3, 4, 5 and 6, allowing repetition, how many unique 6-digit combinations exist?

We need to carefully account for the duplicate digits. There are multiple valid methods.

Method 1 – Using Permutations

1. Calculate total permutations allowing repeats:

   • 6 positions
   • 6 choices for each (1, 2, 3, 4, 5 or 6)
   • So 6^6 potential permutations = 46656

2. Divide by permutations of duplicate digits:

   • 2 spots for 1s: 2! = 2
   • 2 spots for 2s : 2! = 2
   • 3 spots for 3s: 3! = 6

3. Final calculation:
   46656 / (2 2 6) = 720

By compensating for arrangements of duplicates, we have the accurate count!

Method 2 – Using Combinations

1. Choose 2 of the 6 slots for the 1s = C(6,2) = 15
2. Choose 2 of the remaining 4 slots for the 2s = C(4,2) = 6
3. Choose 3 of the remaining 2 slots for the 3s = C(2,3) = 2
4. The last slot gets the 4
5. Multiply totals:
   15 6 2 * 1 = 720

Both methods arrive at the same result – succinctly demonstrating the link between permutations, combinations and counting principles.

Game Design Applications

We can apply similar logic to probability calculations in card games, platformers or RPGs.

Permutation-Based Probability

• Blackjack – probability of being dealt a certain hand

  • Example: Probability of 2-card 20 point hand

    • Permutations with repeats:

      • Choose 1st card: C(13, 1) = 13
      • Choose 2nd card: C(4, 1) = 4

    • Total hands possible:

      • All 2-card hands: C(52,2) = 1326

    • Probability = Specific Hands / All Possible Hands
      = 13 * 4 / 1326
      = 1.5%

• Dice rolls – modeling exact probabilities

  Dice Roll	Probability
  Snake Eyes	(1/6)^2 = 2.8%
  8 Total Pips	C(6,3) (1/6)^3 (5/6)^3 = 5.6%

Combination-Based Game Mechanics

• Passwords/Access Codes – using digit combinations
  • 7-digit entrance code to bank vault!
  • Total codes = 10^7 = 10 million
• Hidden object puzzles – number of clicking combos
  • Find 3 items out of 12 total
  • Solutions: C(12,3) = 220
• Character customizations
  • Color schemes from selection of red, blue, green, purple
    • Shirt/pant combos = C(4,2) = 6
• Procedurally generated rooms from sets of wall/flooring/door assets
  • Calculate permutations for replayability

The possibilities are endless when we understand the foundations!

Conclusions and Next Steps

In this guide, we explored the 720 total combinations possible with the digits 1-6 with repetition. By wielding essential combinatorics – permutations, combinations and factorials – we derived the solution through two elegant methods. We also connected these concepts to probabilities, puzzles and design mechanics we might create or analyze in games.

The intersections of math, programming and design hold endless discovery opportunities. If you have any other game-related math questions or are interested in collaborating, feel free to reach out! Now to brainstorm some new puzzle room designs based on combination-locks…