san2009 wrote:Thanks Mitch. Very helpful!
I had a follow-up question though.
When order does NOT matter - we are supposed to divide by (number of slots)!
Although I understand the reason why...I'm trying to understand that a bit better still
You divided each sub category by the (number of slots)!
Why is that?
Is that always how it is?
I read somewhere that we're supposed to divide by number of things chosen
which is why I thought maybe we are supposed to divide the first scenario (1 fruit 1 cheese) by 2!
and the second scenario (2 fruits 2 cheeses) by 4!.
Thanks again!!
In this problem, we're combining from different sources (our fruit source and our cheese source).
We need to deal with each choice separately.
Since order doesn't matter, to determine the number of combinations possible from each source, we need to divide by (the number of elements being chosen)!:
From our cheese source, we have 6 choices.
A combination of 2 cheeses requires 2 slots, so we divide by 2!: (6*5)/(2*1) = 15.
A combination of 3 cheeses would require 3 slots, so we would divide by 3!: (6*5*4)/(3*2*1) = 20.
A combination of 4 cheeses would require 4 slots, so we would divide by 4!: (6*5*4*3)/(4*3*2*1) = 15.
From our fruit source, we have 2 choices:
A combination of 1 fruit requires 1 slot, so we divide by 1!: 2/1 = 2.
A combination of 2 fruits requires 2 slots, so we divide by 2!: (2*1)/(2*1) = 1.
So to make a plate with 3 cheeses and 2 fruits, we would multiply the results above: 20*1 = 20 possible combinations.
Does this help?
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