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 kec11494 December 18th, 2010 07:26 PM

Counting problem

Hi

I've encountered a counting problem. I think I've calculated the answer but I don't have the solution for checking.

49 balls marked 1, 2, 3, 4, ..., 48, 49
Pick 6 balls from them (49C6 combination, right?)
Assume we have A1, A2, A3, A4, A5 and A6 picked from the 49 balls such that An - An-1 >= 5.
How many combination are there?

First I define a bijection between picked balls and a binary string.
1, 2, 3, 4, 5, ..., 46, 47, 48, 49
For example, if ball 1 is picked, we then use 1 to represent ball 1, 2, 3, 4, 5.
Similarly, if 46 is picked, we use 1 to represent ball 46, 47, 48, 49.
For any balls that are not picked, we use a 0 to represent it.

If ball 46 is one of the six balls, we get (49-(5*5+4))C6 different combination.
If ball 47 is one of the six balls, we get (49-(5*5+3))C6 different combination.
If ball 48 is one of the six balls, we get (49-(5*5+2))C6 different combination.
If ball 49 is one of the six balls, we get (49-(5*5+1))C6 different combination.
If none of 46, 47, 48 or 49 is selected, we get a general case (49-(5*5+5))C6 different combination.

So we sum them all up and get 295716 different combination. Are there any bugs in the calculation?

 dalant019 December 20th, 2010 08:44 PM

Re: Counting problem

If you are familiar with programming, you can write an algorithm with the bijection and your way running in your brain.

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