Question

If a and b are greatest values of ²ⁿC_r and ^{2n - 1}C_r respectively, then
1. a = 2b
2. b = 2a
3. a = b
4. a² = 2b²

Answer 1

Correct Answer - Option 1 : a = 2b

CONCEPT:

We know that the greatest value of ⁿC_r is given by - \(\left\{ {\begin{array}{*{20}{l}} {^n{C_{n/2}}}&{{\rm{if\ n\ is\ even}}}\\ {\frac{{^n{C_{n - 1}}}}{2}\;\:{\rm{or}}\;\:\frac{{^n{C_{n + 1}}}}{2}}&{{\rm{if\ n \ is \ odd}}} \end{array}} \right\}\)

CALCULATIONS:

As given a = Greatest value of ²ⁿC_r = ²ⁿCn and b = greatest value of ^{2n - 1}C_r  = ^{2n - 1}C_{n - 1}

\( \Rightarrow \frac{a}{b} = \frac{{^{2n}{C_n}}}{{^{2n - 1}{C_{n - 1}}}} = \frac{n}{{^{2n - 1}{C_{n - 1}}}} = 2 \Rightarrow a = 2b\)

Therefore option (1) is the correct answer.