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You have a sequence of integers from 1 to 10 and can choose four such that no two are adjacent to each other. How many combinations are there? For example 1,3,5,7 is a good ccombination, but 1,2,6,8 is not.

Can anyone help me with this problem?

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2 Answers 2

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The final outcome is a string of length ten, containing four ones and six zeros. We begin with writing the zeros with spaces between them and at the ends; so there are seven spaces in all. Since each space may contain at most one one we just have to select four of the seven spaces, and this can be done in ${7\choose 4}=35$ ways.

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Think of the 1's as the chosen numbers and the zeros as the numbers that aren't chosen: _0_0_0_0_0_0_. We have seven spaces into which we can put four ones. So the answer is $\binom {7}{4} = 35$

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