If $\mathrm{a}, \mathrm{b}$ and c are the greatest values of ${ }^{19} \mathrm{C}_{\mathrm{p}},{ }^{20} \mathrm{C}_{\mathrm{q}}$ and ${ }^{21} \mathrm{C}_{\mathrm{r}}$ respectively, then:

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

(A) $\frac{\mathrm{a}}{10}=\frac{\mathrm{b}}{11}=\frac{\mathrm{c}}{42}$

(B) $\frac{\mathrm{a}}{11}=\frac{\mathrm{b}}{22}=\frac{\mathrm{c}}{21}$

(D) $\frac{\mathrm{a}}{10}=\frac{\mathrm{b}}{11}=\frac{\mathrm{c}}{21}$

JEE Main · 2020 · 08 Jan 2020 · Shift 1 · Binomial Theorem

a, b

and c are the greatest values of

^{19} C_{p},^{20} C_{q}

and

^{21} C_{r}

respectively, then:

Answer: (C)

The binomial coefficient

^{n} C_{r}

is largest at the central value of

r

. Thus

a =^{19} C_{9} =^{19} C_{10}

b =^{20} C_{10}

and

c =^{21} C_{10} =^{21} C_{11}

. Using

^{n} C_{r} = \frac{n}{r}^{n - 1} C_{r - 1}

b = \frac{20}{10}^{19} C_{9} = 2 a

and

c = \frac{21}{11}^{20} C_{10} = \frac{21}{11} \cdot 2 a = \frac{42 a}{11}

. These relations give

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

JEE Main · 2020 · 08 Jan 2020 · Shift 1 · Single correct option