Question:

<p><strong>Solution:</strong></p><p>Using the identity $\binom{n}{r-1} + \binom{n}{r} = \binom{n+1}{r}$:</p><p>$\binom{18}{r-2} + 2\binom{18}{r-1} + \binom{18}{r} = \binom{18}{r-2} + \binom{18}{r-1} + \left(\binom{18}{r-1} + \binom{18}{r}\right)$</p><p>$= \binom{19}{r-1} + \binom{19}{r}$</p><p>$= \binom{20}{r}$</p><p>We need $\binom{20}{r} \geq 13$.</p><p>Computing values: $\binom{20}{1} = 20, \binom{20}{2} = 190, \ldots$ All values except the extreme values satisfy this.</p><p>Valid values of $r$ are $1, 2, \ldots, 19$ giving $7$ values in the constraint range.</p>