Prove that: ^ 2 n C_n=2^n[1 3 5 (2 n-1)] n! . - Vidyadip

Question

Prove that: ${ }^{2 n} C_n=\frac{2^n[1 \cdot 3 \cdot 5 \ldots \ldots(2 n-1)]}{n!}$.

✓

Answer

${ }^{2 n} C_n=\frac{2^n[1 \cdot 35 \ldots .(2 n-1)]}{n!}$
$=\frac{(2 n)!}{n!n!}$
$(2 n)(2 n-1)(2 n-2)(2 n-3) \ldots . .$
$=\frac{4 \cdot 3 \cdot 2 \cdot 1}{n!n!}$
$=\frac{\left[2^n \ldots .4 \cdot 2\right][(2 n-1) \ldots \cdot 3 \cdot 1]}{n!n!}$
$=\frac{2^n[1 \cdot 2 \cdot \ldots . n][1 \cdot 3 \cdot 5 \ldots \ldots \cdot(2 n-1)]}{n!n!}$
$=\frac{2^n \times n![1 \cdot 3 \cdot 5 \ldots . \cdot(2 n-1)]}{n!n!}$
$=\frac{2^n[1 \cdot 3 \cdot 5 \ldots \cdot(2 n-1)]}{n!}$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "3 Marks Question" from Model Paper 7→Model Paper 7 — all question types→Maths STD 11 Science question papers→CBSE Board STD 11 Science question papers→CBSE Board question paper generator→

Similar questions

Find the equation to the straight line which passes through the point (5, 6) and has intercepts on the axes Equal in magnitude and both positive,Equal in magnitude but opposite in sign.

Differentiate the following functions with respect to x:$\frac{\text{x}}{1+\tan\text{x}}$

Using binomial theorem write down the expansions of the following:
$(1-2\text{x}+3\text{x}^2)^3$

At what point of the parabola $x^2 = 9y$ is the abscissa three times that of ordinate$?$

If P(n) is the statement "$n^2 + n$ is even", and if $P(r)$ is true, then $P(r + 1)$ is true.

Six new employees, two of whom are married to each other, are to be assigned six desks that are lined up in a row. If the assignment of employees to desks is made randomly, what is the probability that the married couple will have nonadjacent desks?
[Hint: First find the probability that the couple has adjacent desks, and then subtract it from 1]

Evaluate:
$\lim\limits_{\text{x} \rightarrow 1}\frac{\text{x}^{7}-2\text{x}^{5}+1}{\text{x}^{3}-3\text{x}^{2}+2}$

Evaluate $\lim\limits_{\text{x}\rightarrow2}{\text{f(x)}}$ (if it exist), where $\text{f(x)}=\begin{cases}\text{x}-[\text{x}],&\text{x}<2\\4 ,& \text{x} = 2\\3\text{x}-5, & \text{x} > 2\end{cases}.$

If $|z|=1$, then prove that $\frac{z-1}{z+1},(z \neq-1)$ is a pure imaginary number. If $z=1$, then what conclusion do you draw from this?

Prove that $\cos\theta\cos\frac{\theta}{2}-\cos3\theta\cos\frac{9\theta}{2}=\sin7\theta\sin8\theta$
$\Big[$Hint: Express $\text{L.H.S.}=\frac{1}{2}\Big[2\cos\theta\cos\frac{\theta}{2}-2\cos3\theta\cos\frac{9\theta}{2}\Big]\Big]$