Combinatorics Question of the Day

[Penetier]

Edit: woops 2^3=8. Disregard if anyone bothered to read.

[Laurabelle]

Edit: woops 2^3=8. Disregard if anyone bothered to read.

This is not correct. For example, suppose raider 1 chooses not to roll, and the 9 others do and win nothing. That's 1 possibility, then count the same thing only raider 2 chooses not to, and so on. There alone are 10 possibilities, which is greater than 8.

[Hagu]

There are 3^10 possible outcomes. Person 1-8 said no, #9 won, #10 lost is an outcome and 1-8 no, #9 lost, #10 won is another.

But there are only 66 i.e. 11*(11+1)/2 different results.

#1 = 10 people no
#2 = 9 no, 1 loss
#3 = 9 no, 1 win
...
#7 = 7 no, 3 loss, 0 win
...
#10 = 7 no, 0 loss, 3 win
...
#65 = 0 no, 1 loss, 9 win
#66 = 0 no, 0 loss, 10 win

[Laurabelle]

There are 3^10 possible outcomes. Person 1-8 said no, #9 won, #10 lost is an outcome and 1-8 no, #9 lost, #10 won is another.

But there are only 66 i.e. 11*(11+1)/2 different results.

#1 = 10 people no
#2 = 9 no, 1 loss
#3 = 9 no, 1 win
...
#7 = 7 no, 3 loss, 0 win
...
#10 = 7 no, 0 loss, 3 win
...
#65 = 0 no, 1 loss, 9 win
#66 = 0 no, 0 loss, 10 win

Depends how you define 'results', as say your case #2, you lost the information by saying which of the 10 was a loss. So you collapse a whole series of outcomes into one category.

[darkwarrior42]

Depends how you define 'results', as say your case #2, you lost the information by saying which of the 10 was a loss. So you collapse a whole series of outcomes into one category.

That's just the difference between a permutation and a combination. You never did specify in the original question whether we care *which* raiders specifically used the coin, and then lost or won... so unless you say otherwise, I'm assuming it's irrelevant (since we don't know who the raiders are, why do we care whether raider A or raider B is the one who use a coin and won?)

[Laurabelle]

Well, typically we consider people as distinct individuals. Either way, both your answers are correct for each case.

[Dole4011]

I think 59049 needs some deductions.

For instance, if "some" of those raiders will use a bonus roll, then doesn't that mean that all 10 opting out of using a bonus roll is not an option (since it would mean none, not some, used a bonus roll).

Also, saying "of those that use a bonus roll some will also win an item from it" suggests to me that at least 1 (arguably 2+) of the bonus rollers must win an item.

0 or none < one < some < all

I may just be being too nit-picky, but I think a more precise way to state the prompt would be along the lines of, "10 raiders kill a raid boss. Any number of the raiders may use a bonus roll, and of those that use a bonus roll any number of them may win an item from it. In how many different ways can this happen?"