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6.Permutation and Combination
hard
यदि $a , b$ तथा $c$ क्रमश: ${ }^{19} C _{ p },{ }^{20} C _{ q }$ तथा ${ }^{21} C _{ r }$ के अधिकतम मान हैं, तो
A
$\frac{a}{11}=\frac{b}{22}=\frac{c}{21}$
B
$\frac{\mathrm{a}}{10}=\frac{\mathrm{b}}{11}=\frac{\mathrm{c}}{21}$
C
$\frac{\mathrm{a}}{10}=\frac{\mathrm{b}}{11}=\frac{\mathrm{c}}{42}$
D
$\frac{a}{11}=\frac{b}{22}=\frac{c}{42}$
(JEE MAIN-2020)
Solution
$a=^{19} \mathrm{C}_{10}, \mathrm{b}=^{20} \mathrm{C}_{10}$ and $\mathrm{c}= ^{21} \mathrm{C}_{10}$
$\Rightarrow \mathrm{a}=^{19} \mathrm{C}_{9}, \mathrm{b}=2\left(^{19} \mathrm{C}_{9}\right)$ and $\mathrm{c}=\frac{21}{11}\left(^{20} \mathrm{C}_{10}\right)$
$\Rightarrow \mathrm{b}=2 \mathrm{a}$ and $\mathrm{c}=\frac{21}{11} \mathrm{b}=\frac{42 \mathrm{a}}{11}$
$\Rightarrow a: b: c=a: 2 a: \frac{42 a}{11}=11: 22: 42$
Standard 11
Mathematics