Re: HSC 2013 4U Marathon
Sounds good
Hm ok... but what about the arrangement (3 grapes, 3 apples)? This is an arrangement for 6 fruits, and it's not proportional to any arrangement of 3 fruits , but yet it isn't legal...
You are almost there
go deeper!
This problem only arises with choosing an even number of fruit, we will disregard the the situation with 6 fruits as you have mentioned as it will be reflected in 12 fruits and hence cancels out the arrangements anyway (but I get there is a problem).
However the problem only arises for even numbers,
that when we half it we get an odd number, so in this case, it will be 2, 6, 10.
We already disregard 2 so that doesnt matter, and we disregard 6 since it cancels out with arrangements with 12 anyway.
Look at 10 fruits being chosen, we can split it into 5 G, 5 A and no arrangement of 5 fruits can reflect this proportion. Hence we disregard that arrangement,
Hence 5726 - 5C2
But there is more to be excluded, we now look at numbers of fruits, such that we can separate them into groups of 3. i.e. multiples of 3:
3, 6, 9, 12
3, 6, 9 disregarded (odd numbers)
Look at 12: We can seperate it into 4 groups of 3, and no arrangement of 6 fruits can reflect this proportionality.
Hence 5726 - 5C2 - 5C4
Now lets check multiples of 4: 4, 8, 12
For 8: we can split into 2 groups of 4, however we can replicate this with 4, and same thing for 12, we can replicate this with 6. The reason being is that for 8, we seperate into 2 groups and 2 is a factor of 4 (whereas earlier 2 is not a factor of 5), and similarly for 12, 3 is a factor of 6.
Check multiples of 5: 5, 10
We already covered this. So the final answer which I hope to be right is:
5711