If _ k =1 ^ 31 ( ^ 31 C _ k ) ( ^ 31 C _ k -1 )- _ k =1 ^ 30 ( ^ … - Vidyadip

Question

If $\sum\limits_{ k =1}^{31}\left({ }^{31} C _{ k }\right)\left({ }^{31} C _{ k -1}\right)-\sum\limits_{ k =1}^{30}\left({ }^{30} C _{ k }\right)\left({ }^{30} C _{ k -1}\right)=\frac{\alpha(60 !)}{(30 !)(31 !)}$

Where $\alpha \in R$, then the value of $16 \alpha$ is equal to

✓

Answer

a
$\sum\limits_{ R =1}^{31}{ }^{31} C _{ R } \cdot{ }^{31} C _{ R -1}$

$={ }^{31} C _{1} \cdot{ }^{31} C _{0}+{ }^{31} C _{2} \cdot{ }^{31} C _{1}+\ldots .+{ }^{31} C _{31} \cdot{ }^{31} C _{30}$

$={ }^{31} C _{0} \cdot{ }^{31} C _{30}+{ }^{31} C _{1} \cdot{ }^{31} C _{29}+\ldots .+{ }^{31} C _{30} \cdot{ }^{31} C _{0}$

$={ }^{62} C _{30} \cdot$

Similarly

$\sum\limits_{ R =1}^{30}\left({ }^{30} C _{ R } \cdot{ }^{30} C _{ R -1}\right)={ }^{60} C _{29}$

${ }^{62} C _{30}-{ }^{60} C _{29}=\frac{62 !}{30 ! 32 !}-\frac{60 !}{29 ! 31 !}$

$=\frac{60 !}{29 ! 31 !}\left\{\frac{62 \cdot 61}{30 \cdot 32}-1\right\}$

$=\frac{60 !}{30 ! 31 !}\left(\frac{2822}{32}\right)$

$\therefore 16 \alpha=16 \times \frac{2822}{32}=1411$

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

If the solution curve of the differential equation $\left(y-2 \log _e x\right) d x+\left(x \log _e x^2\right) d y=0, x > 1$ passes through the points $\left(e, \frac{4}{3}\right)$ and $\left(e^4, \alpha\right)$, then $\alpha$ is equal to $................$.

The value of $\left| {\,\begin{array}{*{20}{c}}{265}&{240}&{219}\\{240}&{225}&{198}\\{219}&{198}&{181}\end{array}\,} \right|$ is equal to

In how many ways can Rs. $16$ be divided into $4$ person when none of them get less than Rs. $3$

Let $A_{1}=\left\{(x, y):|x| \leq y^{2},|x|+2 y \leq 8\right\}$ and $A_{2}=\{(x, y):|x|+|y| \leq k\}$. If $27$ (Area $\left.A _{1}\right)=5$ (Area $A _{2}$ ), then $k$ is equal to

Let the tangent to the parabola $y^2=12 x$ at the point $(3, \alpha)$ be perpendicular to the line $2 x+2 y=3$.Then the square of distance of the point $(6,-4)$from the normal to the hyperbola $\alpha^2 x^2-9 y^2=9 \alpha^2$at its point $(\alpha-1, \alpha+2)$ is equal to $........$.

The sum of two non-zero numbers is $4$ . The minimum value of the sum of their reciprocals is

A water tank has the shape of a right circular cone with axis vertical and vertex downwards. Its semivertical angle is $\tan ^{-1} \frac{3}{4}$. Water is poured in it at a constant rate of $6$ cubic meter per hour. The rate (in square meter per hour), at which the wet curved surface area of the tank is increasing, when the depth of water in the tank is $4$ meters, is.

If the mean of the data : $7, 8, 9, 7, 8, 7, \mathop \lambda \limits^. , 8$ is $8$, then the variance of this data is

The variance of $\alpha$, $\beta$ and $\gamma$ is $9$, then variance of $5$$\alpha$, $5$$\beta$ and $5$$\gamma$ is

Let $\vec{a}=\hat{i}+2 \hat{j}+\lambda \hat{k}, \vec{b}=3 \hat{i}-5 \hat{j}-\lambda \hat{k}, \vec{a} \cdot \vec{c}=7$, $2 \vec{b} \cdot \vec{c}+43=0, \vec{a} \times \vec{c}=\vec{b} \times \vec{c}$. Then $|\vec{a} \cdot \vec{b}|$ is equal to