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| Math Central | Quandaries & Queries |
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Question from Meghan, a student: Sorry to ask another question so soon! How many combinations are there of 3 numbers chosen from 1, 2, 3, 4, 5, 6, 7, 8, 9 and 10 but where no 2 of the 3 numbers are consecutive? Thank you! |
Meghan,
you first need to count the total numbers of ways to pick 3 numbers (10C3) and then subtract off that the number of 'bad' combinations that have two consecutive numbers. This last part is a bit tricky. One approach is to first pick two consecutive numbers (in how many ways?) and then pick a 3rd number from the rest; the result is a combination with at least two consecutive numbers. But, for example, if you picked 45 say, and then picked 6 you'll notice that you end up with the same 'bad' triple as if you'd first picked 56 and then picked the 4. So, you can easily end up counting the same triple twice and we don't want to do that. How often does this happen? When you have 3 consecutive numbers as your triple. How many of these are there?
Thus you need:
[total number of triples] -( [triples with at least 2 consecutive] - [triples with 3 consecutive]) =
[total number of triples] - [triples with at least 2 consecutive] + [triples with 3 consecutive].
Penny
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