If (18r−2)+2(18r−1)+(18r)≥(2013) then the number of values of r are

Cr−2 18+2⋅Cr−1 18+Cr 18≥C13 20(Cr−2 18+Cr−1 18)+(Cr−1 18+Cr 18)≥C13 20Cr−1 19+Cr 19≥C13 20Cr 20≥C13 20 Cr 20≥C7 20 ∴ 7≤r≤13 r=7, 8, 9, 10, 11, 12, 13 No of values of r are 7

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If $(\begin{matrix} 18 \\ r - 2 \end{matrix}) + 2 (\begin{matrix} 18 \\ r - 1 \end{matrix}) + (\begin{matrix} 18 \\ r \end{matrix}) \geq (\begin{matrix} 20 \\ 13 \end{matrix})$ then the number of values of r are

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answer is 7.

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Detailed Solution

${}^{18}C_{r - 2} + 2 \cdot {}^{18}C_{r - 1} + {}^{18}C_{r} \geq {}^{20}C_{13}$ $({}^{18}C_{r - 2} + {}^{18}C_{r - 1}) + ({}^{18}C_{r - 1} + {}^{18}C_{r}) \geq {}^{20}C_{13}$