Three different integers from 1 to 11 are selected. In how many of these combinations of 3 numbers are their sums a multiple of 3?

- pythagoras123

- jamiebookeater

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- pythagoras123

how did you get 26? O_O

- pythagoras123

erm... no.

- inkyvoyd

Gah, FFM, you just gave me another strange problem to think about. The sum of the fib numbers?

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## More answers

- dumbcow

the answer is 57 but i don't have a mathematical proof for you
i used a computer to evaluate all possible cases

- anonymous

@FoolForMath... Can I ask what combination of 3numbers is. does that mean order doesn't matter? or for instance, in this cause would 132 be same as 123?

- anonymous

Btw what is the problem source?

- pythagoras123

Homework

- pythagoras123

That's why I need an explanation not just an answer

- anonymous

Competitive math probably IMO?

- pythagoras123

dunno, but looks like it

- dumbcow

the distribution is as follows for combinations that sum up to 6,9,12 ...
1+3+7+11+13+11+7+3+1

- anonymous

@anonymoustwo44: As this is combination 1+2+3 is same as 2+1+3

- anonymous

you can try like that:
the sum of numbers a,b,c is divisible by 3 if the sum of their digits is divisible by 3.

- anonymous

@FoolForMath so the number 231 will be the same as the combination 321?

- anonymous

they bouth divisible by 3

- anonymous

so are they the same combination?

- anonymous

what do you mean?
the maximum sum to check in this case is: 11+10+9 = 30 (digits sum =3)
you never even get to 123 or 321

- anonymous

never mind me :))

- anonymous

Every number can be expressed as 3n, 3n+1,3n+2
Case1. 3n,3n,3n
Case2. 3n, 3n+1,3n+2
Case3. 3n+2,3n+2,3n+2
Case 4. 3n+1,3n+1,3n+1
In the set {1,2,..,11} 3 (3n forms), 4-4 (3n+1 and 3n+2)
3c3 + 3c1*4c1*4c1 + 4c3 + 4c1
How about this? :D

- anonymous

3c3 + 3c1*4c1*4c1 + 4c3 + 4c3

- anonymous

did i miss something? some case?maybe

- anonymous

myko?

- anonymous

just you have to be carefull with n=0.... wait checking...

- anonymous

i only counted 3, for 3n forms {3,6,9}

- anonymous

3n+1
3n+2
for n = 0
1 and 2

- anonymous

never mind, you mean other thing, sry

- anonymous

you right, good job

- anonymous

Ishaan is correct

- anonymous

Answer for this problem is 26

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