Q101 / 180 · KCET 2014 · Mathematics
If \( 21^{\text {st }} \) and \( 22^{\text {nd }} \) terms in the expansion of \( (1+x)^{44} \) are equal, then \( x \) is equal to
- A\( \frac{21}{22} \)
- B\( \frac{23}{24} \)
- C\( \frac{8}{7} \)
- Correct answerD\( \frac{7}{8} \)
Solution
Given that \( (1+x)^{44} \)
\[ \begin{array}{l} \text { Now, }{ }^{44} C_{20} x^{20}={ }^{44} C_{21} x^{21} \\ \Rightarrow x=\frac{{ }^{44} C_{20}}{{ }^{44} C_{2}} \\ =\frac{(44-21) ! 21 !}{(44-20) ! 20 !}=\frac{23 !}{24 !} \cdot \frac{21 !}{20 !} \\ =\frac{21}{24}=\frac{7}{8} \end{array} \]
\[ \begin{array}{l} \text { Now, }{ }^{44} C_{20} x^{20}={ }^{44} C_{21} x^{21} \\ \Rightarrow x=\frac{{ }^{44} C_{20}}{{ }^{44} C_{2}} \\ =\frac{(44-21) ! 21 !}{(44-20) ! 20 !}=\frac{23 !}{24 !} \cdot \frac{21 !}{20 !} \\ =\frac{21}{24}=\frac{7}{8} \end{array} \]
Hence, the correct answer is option: 'D'.