Question 17
Let $$S=\{1,2,3,\ldots,15\}$$. The number of four-element subsets $$A$$ of $$S$$ such that any two elements of $$A$$ differ by at least $$2$$ is
Correct Answer: 495
Solution
Write the selected elements as $$x_1<x_2<x_3<x_4$$ with $$x_{i+1}-x_i\geq2$$. Define $$y_i=x_i-(i-1)$$, which gives $$1\leq y_1<y_2<y_3<y_4\leq12$$. Therefore, the number of choices is $$\binom{12}{4}=495$$.
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