Question:
The greatest coefficient in the expansion of \(\left(x + \dfrac{1}{x}\right)^{2n}\) is
\(\left(x + \dfrac{1}{x}\right)^{2n}\) के प्रसार में महत्तम गुणांक क्या है?
Options:
- \(\dfrac{2!}{(n!)^2}\)
- \(\dfrac{n!}{\left[\left(\frac{n}{2}\right)!\right]^2}\)
- \(\dfrac{1 \cdot 3 \cdot 5 \cdots (2n-1) \cdot 2^n}{n!}\)
- \(\dfrac{2!}{[(n!)^2]}\)
- \(\dfrac{2!}{(n!)^2}\)
- \(\dfrac{n!}{\left[\left(\frac{n}{2}\right)!\right]^2}\)
- \(\dfrac{1 \cdot 3 \cdot 5 \cdots (2n-1) \cdot 2^n}{n!}\)
- \(\dfrac{2!}{[(n!)^2]}\)
Correct Answer: c)
Explanation:
The greatest coefficient in the expansion of \(\left(x + \dfrac{1}{x}\right)^{2n}\) is the middle term coefficient \({}^{2n}C_n\).
\({}^{2n}C_n = \dfrac{(2n)!}{(n!)^2} = \dfrac{1 \cdot 3 \cdot 5 \cdots (2n-1) \cdot 2^n \cdot n!}{(n!)^2} = \dfrac{1 \cdot 3 \cdot 5 \cdots (2n-1) \cdot 2^n}{n!}\).
\(\left(x + \dfrac{1}{x}\right)^{2n}\) के प्रसार में महत्तम गुणांक मध्य पद का गुणांक \({}^{2n}C_n\) होता है।
\({}^{2n}C_n = \dfrac{(2n)!}{(n!)^2} = \dfrac{1 \cdot 3 \cdot 5 \cdots (2n-1) \cdot 2^n}{n!}\)।