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# Combinatorics question Watch

1. At the start of a player's turn in a game of RISK, they have a territories and are granted d reinforcements. (a and d being positive integers)

They then allocate reinforcements one-by-one to any of their territories. They may place 1 reinforcement on each of d territories (assuming a>d), or all d reinforcements on a single territory, or any other allocation of the reinforcements, as they wish.

How do I calculate the number of possible allocations of reinforcements for that player in terms of a and d?

Edit: I'd think this would be to do with permutations rather than combinations, because all that matters is the final result of allocation of reinforcements; the order of where reinforcements are placed is irrelevant.
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So, you're trying to divide up "d" units, into "a" different territories.

Lets start by assigning the territories an order (any order will do). We have, in effect created "a" slots.

If we represent a unit by "0", and a divider between adjacent slots by "|", then for example allocating 3 units to 3 slots could be represented one way 000||, i.e. they all go to the first slot (territory), or 0|0|0, and one goes to each territory.

Given the previous questions you've asked on combinatorics, can you now see how to proceed?
3. (Original post by ghostwalker)
So, you're trying to divide up "a" units, into "d" different territories.

Lets start by assigning the territories an order (any order will do). We have, in effect created "d" slots.

If we represent a unit by "0", and a divider between adjacent slots by "|", then for example allocating 3 units to 2 slots could be represented one way 000||, i.e. they all go to the first slot (territory), or 0|0|0, and one goes to each territory.

Given the previous questions you've asked on combinatorics, can you now see how to proceed?
Ah I see - so it just looks like a simple case of combinations with repetitions now - so (a+d-1)! / ( d! * (a-1)! ) is right?
Ah I see - so it just looks like a simple case of combinations with repetitions now - so (a+d-1)! / ( d! * (a-1)! ) is right?
Yep.

Sorry, I had a and d allocated the other way around. "a" is the number of territories.

Edit: Corrected my previous post.

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