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

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

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→Rajasthan Board STD 11 Science question papers→Rajasthan Board question paper generator→

Similar questions

$\frac{\text{n!}}{(\text{n-r})!}= \text{n}(\text{n}-1)(\text{n}-2).....(\text{n}(\text{n}-\text{r}))$

Prove that the line y -x + 2 = 0 divides the join of points (3, -1) and (8, 9) in the ratio 2 : 3.

Show that the products of the corresponding terms of the sequences a, ar, ar² ,........arⁿ^-1 and A, AR, AR²......AR^n-1form a G.P. and find the common ratio.

The sum of three numbers a, b, c in A.P. is 18. If a and b are each increased by 4 and c is increased by 36, the new numbers from a G.P. Find a, b, c.

Find the general solutions of the following equations:
$\sin\text{x}=\tan\text{x}$

Find the values of the other five trigonometric functions in the following:
$\cot\text{x}=-\frac{1}{2},$ x in quadrant II

The longest side of a triangle is three times the shortest side and third side is 2 cm shorter than the longest side if the perimeter of the triangles at least 61 cm, find the minimum length of the shortest-side.

Evaluate the following limit:
$\lim\limits_{\text{x}\rightarrow\sqrt{2}}\frac{\text{x}^2-2}{{\text{x}^2+\sqrt{2}\text{x}-4}}$

Evaluate the following limit:
$\lim\limits_{\text{x}\rightarrow0}\frac{\text{x}\tan\text{x}}{1-\cos\text{x}}$

Find the derivative of $x^{-4}\left(3-4 x^{-5}\right)$