The number of value of in set of value of $r$ for which ${^{23} C}_{r} + 2. {^{23} C}_{r + 1} + {^{23} C}_{r + 2} \geq {^{25} C}_{15}$ is

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Solution

We know $^{n} C_{r} +^{n} C_{r - 1} =^{n + 1} C_{r}$
$^{23} C_{r} +^{23} C_{r + 1} +^{23} C_{r + 1} +^{23} C_{r + 2} ⩾^{25} C_{15}$
$\Rightarrow^{23} C_{r + 1} +^{23} C_{r} +^{23} C_{r + 2} +^{23} C_{r + 1} ⩾^{25} C_{15}$
$\Rightarrow^{24} C_{r + 1} +^{24} C_{r + 2} ⩾^{25} C_{15}$
$\Rightarrow^{24} C_{r + 2} +^{24} C_{r + 1} ⩾^{25} C_{15}$
$\Rightarrow^{25} C_{r + 2} ⩾^{25} C_{15}$
The Inequity holds when
$r = 8, 9, 10, 11, 12, 13$
Hence, number of value of 'r' which satisfies
the given Equation is '6'.

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