# If a, b and c are the greatest values of $^{19}C_{p},^{20}C_{q},^{21}C_{r}$ respectively, then:   Option: 1 Option: 2 Option: 3 Option: 4

Binomial Coefficient of the middle term is greatest.

Now,

$\\^nC_{r}\;\text{ is max at middle term}\\\begin{array}{l}{a=^{19} C_{p}=^{19} C_{10}=^{19} C_{9}} \\ {b=^{20} C_{q}=^{20} C_{10}} \\ {c=^{21} C_{r}=^{21} C_{10}=^{21} C_{11}}\end{array}$

$\frac{a}{^{19}C_9}=\frac{b}{ ^{20} \mathrm{C}_{10}}=\frac{c}{^{21} \mathrm{C}_{11}}$

$\frac{a}{^{19}C_9}=\frac{b}{\frac{20}{10} \cdot ^{19} \mathrm{C}_9}=\frac{c}{\frac{21}{11} \cdot \frac{20}{10} ^{19} \mathrm{C}_{9}}$

$\\\frac{a}{1}=\frac{b}{2}=\frac{c}{42 / 11}\\\frac{a}{11}=\frac{b}{22}=\frac{c}{42}$

Correct Option (1)

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