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Three numbers are lo be selected from the set of the first 15 natural numbers so that all the selected ones are consecutive or none of them are consecutive. The number of ways that selection can be made is

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286

299

300

312

answer is B.

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Detailed Solution

Consecutive numbers $1 - 3, 2 - 4, 3 - 5, \dots$

$13 - 15, 13$ in all

The number of ways that no two are consecutive is

$(\begin{matrix} 15 - 3 + 1 \\ 3 \end{matrix}) = (\begin{matrix} 13 \\ 3 \end{matrix}) = 286$

the desired number is $13 + 286 = 299$

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