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Finding maximum number of combinations which include specific numbers


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Let's say I want to group numbers between 1 and 50 with 4 columns:

1 2 3 4
1 2 3 5
1 2 3 6
...
2 3 4 5
2 3 4 6
...
47 48 49 50



To find the total possible combinations,we do:
50*49*48*47
__________
1*2*3*4
This formula gives the maximum possible combinations.


My question is,what would the formula be to find maximum of how many combinations of
these numbers include number 1 and 2(for example)?

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You would fix the specific numbers in the first however many slots and then work out the number of combinations for the remaining slots.

 

In your initial example of finding all groups, you're calculating [math]50\choose4[/math] to find the number of possible combinations of four integers between 1 and 50 inclusive. If you wanted to find all combinations containing 1 and 2, then you would assume 1 and 2 occupy two slots, and then calculate how many ways to choose two numbers from the remaining 48 possibilities, i.e. you'd calculate [math]{{48}\choose{2}} = \frac{48!}{2!(48-2)!} = \frac{48\times47}{2}[/math].

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