If a, b and c are the greatest values of 19Cp, 20Cq and 21Cr respectively, then :

Engineering

Mathematics

Properties of Binomial Coefficients

Question

If a, b and c are the greatest values of ¹⁹C_p, ²⁰C_q and ²¹C_r respectively, then :

$\frac{a}{10} = \frac{b}{11} = \frac{c}{21}$

$\frac{a}{11} = \frac{b}{22} = \frac{c}{21}$

$\frac{a}{10} = \frac{b}{11} = \frac{c}{42}$

$\frac{a}{11} = \frac{b}{22} = \frac{c}{42}$

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We know ⁿC_r is max at middle term

a = ¹⁹C_p = ¹⁹C₁₀ = ¹⁹C₉

b = ²⁰C_q = ²⁰C₁₀

c = ²¹C₆ = ²¹C₁₀ = ²¹C₁₁

$\frac{a}{^{19} C_{9}} = \frac{b}{\frac{20}{10} .^{19} C_{9}} = \frac{c}{\frac{21}{11} . \frac{20}{10}^{19} C_{9}}$

$\frac{a}{1} = \frac{b}{2} = \frac{c}{42 / 11}$

$\frac{a}{11} = \frac{b}{22} = \frac{c}{42}$