Binomial Theorem Question - None

Question

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

Accepted Answer

Solution: The greatest binomial coefficient in $\binom{n}{r}$ occurs at the middle term.

For $n = 19$ (odd): $a = \max\binom{19}{p} = \binom{19}{9} = \binom{19}{10}$ — there are two equal maximum values

For $n = 20$ (even): $b = \max\binom{20}{q} = \binom{20}{10}$

For $n = 21$ (odd): $c = \max\binom{21}{r} = \binom{21}{10} = \binom{21}{11}$ — there are two equal maximum values

Then: $\frac{a}{\binom{19}{10}} = 1$, $\frac{b}{\binom{20}{10}} = 1$, $\frac{c}{\binom{21}{10}} = 1$

Computing the ratio: $\frac{\binom{19}{10}}{\binom{20}{10}} = \frac{11}{20}$ and $\frac{\binom{20}{10}}{\binom{21}{10}} = \frac{11}{21}$

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

∴ Answer is (D).

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