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How many sets of three integers between 1 and 20 are possible if no two consecutive
integers are to be in a set?

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It shouldnt be consecutive ,.... so we can divide the numbers into 2 parts containing 10 values each

one part contains even numbers and other part contains odd numbers

by this, in the two parts we dont have numbers that are consecutive

now forming sets of threee numbers from these two parts we get

10C3 ways from part 1 and 10C3 ways from part 2

so 10C3 * 10C3 =14400

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