First slide

Combinations

Question

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

Moderate

A

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

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

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

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

Solution

We know ${}^{n}C_{\frac{n}{2}}$ (if n is even) and ${}^{n}C_{\frac{n - 1}{2}} o r {}^{n}C_{\frac{n + 1}{2}}$ (if n is odd) are greatest at middle term(s)
$∴$ $a = {}^{19}C_{p} = {}^{19}C_{10} = {}^{19}C_{9}$
and $b = {}^{20}C_{q} = {}^{20}C_{10}$
and $c = {}^{21}C_{r} = {}^{21}C_{10} = {}^{21}C_{11}$
Now, using $\frac{{}^{n}C_{r}}{{}^{n - 1}C_{r - 1}} = \frac{n}{r}$ , we get $\frac{a}{{}^{19}C_{9}} = \frac{b}{\frac{20}{10} . {}^{19}C_{9}} = \frac{c}{\frac{21}{11} . \frac{20}{10} {}^{19}C_{9}}$
$\Rightarrow \frac{a}{1} = \frac{b}{2} = \frac{c}{\frac{42}{11}}$ $\Rightarrow \frac{a}{11} = \frac{b}{22} = \frac{c}{42}$

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