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  4. The sum of values of r for which 18 Cr-2+2 ⋅ 18 Cr-1+ 18 Cr ≥ 20 C13

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Q. The sum of values of $r$ for which ${ }^{18} C_{r-2}+2 \cdot{ }^{18} C_{r-1}+{ }^{18} C_r \geq{ }^{20} C_{13}$

NTA AbhyasNTA Abhyas 2022

Solution:

Using formula ${ }^n C_r+{ }^n C_{r-1}={ }^{n+1} C_r $
${ }^{18} C_{r-2}+{ }^{18} C_{r-1}+{ }^{18} C_{r-1}+{ }^{18} C_r \geq{ }^{20} C_{13} $
${ }^{19} C_{r-1}+{ }^{19} C_r \geq{ }^{20} C_{13} $
${ }^{20} C_r \geq{ }^{20} C_{13} \text { or }{ }^{20} C_{20-r} \geq{ }^{20} C_{13}$
$r=7,8,9,10,11,12,13$
Sum of all values of $r=70$.