If ( ^ 40 C _ 0 )+ ( ^ 41 C _ 1 )+ ( ^ 42 C _ 2 )+ + ( ^ C _ 20 )… - Vidyadip

MCQ

If $\left({ }^{40} C _{0}\right)+\left({ }^{41} C _{1}\right)+\left({ }^{42} C _{2}\right)+\ldots+\left({ }^{\infty} C _{20}\right)=\frac{ m }{ n }{ }^{60} C _{20}, m$ and $n$ are coprime, then $m+n$ is equal to

✓
$102$
B
$103$
C
$104$
D
$105$

✓

Answer

Correct option: A.

$102$

a
${ }^{40} C _{0}+{ }^{41} C _{1}+{ }^{42} C _{2}+\ldots . .{ }^{59} C _{19}+{ }^{60} C _{20}$

$\left(\frac{1}{41}+1\right){ }^{41} C _{1}+{ }^{42} C _{2}+\ldots \ldots$

$\left[\frac{42}{41}\left(\frac{2}{42}\right)+1\right]{ }^{42} C _{2}+{ }^{43} C _{3}+\ldots .$

$\left(\frac{2}{41}+1\right)^{42} C _{2}+{ }^{43} C _{3}+\ldots . .$

$\left(\frac{43}{41} \times \frac{3}{43}+1\right){ }^{43} C _{3}+{ }^{44} C _{4}+\ldots \ldots .$

$\frac{3+41}{41}{ }^{43} C _{3}+\ldots \ldots .$

Similarly :

$\frac{20+41}{41}$

$\Rightarrow m =61 ; n =41$

$m + n =102$

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

JEE STD 11 Science question papers→Gujarat Board STD 11 Science question papers→Gujarat Board question paper generator→

Similar questions

Let $f: R \rightarrow R$ be defined as $f(x)=x^{4} .$ Choose the correct answer.

If $\frac{\sqrt{2} \sin \alpha}{\sqrt{1+\cos 2 \alpha}}=\frac{1}{7}$ and $\sqrt{\frac{1-\cos 2 \beta}{2}}=\frac{1}{\sqrt{10}}$ $\alpha, \beta \in\left(0, \frac{\pi}{2}\right),$ then $\tan (\alpha+2 \beta)$ is equal to

If $u = {e^{ - {x^2} - {y^2}}}$, then

Which is the correct order for a given number $\alpha $in increasing order

For $0 < x < \frac{\pi }{2},\int\limits_{\frac{1}{2}}^{\frac{{\sqrt 3 }}{2}} {} $ ln $(e^{cos x})$. $d (sin\, x)$ is equal to :

Let $z_1, z_2$ and $z_3$ be three complex numbers on the circle $|z|=1$ with $\arg \left(z_1\right)=\frac{-\pi}{4}, \arg \left(z_2\right)=0$ and $\arg \left(z_3\right)=\frac{\pi}{4}$. If $\left|z_1 \bar{z}_2+z_2 \bar{z}_3+z_3 \bar{z}_1\right|^2=\alpha+\beta \sqrt{2}, \alpha, \beta \in Z$, then the value of $\alpha^2+\beta^2$ is :

The centre of a circle $C$ is at the centre of the ellipse $E: \frac{x^2}{a^2}+\frac{y^2}{b^2}=1, a>b$. Let $C$ pass through the foci $F_1$ and $F_2$ of $E$ such that the circle $C$ and the ellipse $E$ intersect at four points. Let P be one of these four points. If the area of the triangle $PF _1 F_2$ is 30 and the length of the major axis of E is 17 , then the distance between the foci of $E$ is :

The value of $\mathop {\lim }\limits_{n \to \infty } \frac{{{x^n}}}{{{x^n} + 1}}$ where $x < - 1$ is

The area bounded by the parabola $y^2 = 4x$ and the line $2x - 3y + 4 = 0$, in square unit, is

Let $f(x)=7 \tan ^8 x+7 \tan ^6 x-3 \tan ^4 x-3 \tan ^2 x$ for all $x \in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$. Then the correct expression$(s)$ is(are)

$(A)$ $\int^{\pi / 4} x f(x) d x=\frac{1}{12}$

$(B)$ $\int_0^{\pi / 4} f(x) d x=0$

$(C)$ $\int_0^{\pi / 4} x f(x) d x=\frac{1}{6}$

$(D)$ $\int_0^{\pi / 4} f(x) d x=1$