Permutations an...

A person always prefers to eat parantha and vegetable dish in his meal. How many ways can he make his plate in a marriage party if there are three types of paranthas, four types of vegetable dishes, three types of salads, and two types of sauces?

Evaluate 
$$^{47}C_4 + \displaystyle\sum_{j=0}^{3}{^{50-j}C_3} + \sum_{k=0}^{5}{^{56-k}C_{53-k}}$$

For $$2\le r \le n$$, $$\left(\begin{matrix}n \\ r\end{matrix}\right) + 2\left(\begin{matrix}n \\ r-1 \end{matrix}\right) + \left(\begin{matrix}n \\ r - 2 \end{matrix}\right) \space =$$

For any positive integer $$m,\space n \space(\mbox{ with } n\ge m) = ^nC_m$$. 
$$\left(\begin{matrix}n \\ m\end{matrix}\right) + \left(\begin{matrix}n - 1 \\ m\end{matrix}\right) + \left(\begin{matrix}n - 2 \\ m\end{matrix}\right) + ... + \left(\begin{matrix}m \\ m\end{matrix}\right) = $$

The number of positive integers satisfying the inequality
$$\quad ^{n+1}C_{n-2} - ^{n+1}C_{n-1} \le 100$$ is

The straight lines $$I_1, I_2, I_3$$ are parallel and lie in the same plane. A total number of $$m$$ points on $$I_1$$; $$n$$ points on $$I_2$$; $$k$$ points on $$I_3$$, the maximum number of triangles formed with vertices at these points are

If $$r, s$$ and $$t$$ are prime numbers and $$p, q$$ are positive integers such that the LCM of $$p,q$$ is $$\displaystyle r^{2}t^{4}s^{2}$$ then the number of ordered pair $$(p, q)$$ is 

If $$\displaystyle ^{7}C_{r} + 3 ^{7}C_{r+1} + 3 ^{7}C_{r+2} + ^{7}C_{r+3} > ^{10}C_{4}$$, then the quadratic equation whose roots are $$\displaystyle \alpha, \: \beta$$ and $$\displaystyle \alpha^{r-1}, \: \beta^{r-1}$$ have

In the figure,two 4-digit numbers are to be formed by filling the places with digits. The number of different ways in which the places can be filled by digits so that the sum of the numbers formed is also a 4-digit number and in no place the addition is with carrying, is

If $$^nC_{r-1}=36, ^nC_r=84$$ and $$^nC_{r+1}=126$$, then r is