The sum $\sum_{r=1}^{10} (r^2 + 1) \times (r!)$ is equal to

$A$ box contains two white balls,three black balls,and four red balls. In how many ways can three balls be drawn from the box if at least one black ball is to be included in the draw?

$^{14}C_4 + \sum_{j=1}^4 {^{18-j}C_3}$ is equal to

If all the words,with or without meaning,are written using the letters of the word $QUEEN$ and are arranged as in an English dictionary,then the position of the word $QUEEN$ is (in $^{th}$)

On the occasion of the Deepawali festival,each student of a class sends greeting cards to all the others. If there are $20$ students in the class,then the total number of greeting cards exchanged by the students is:

In an examination hall,there are $4$ rows of chairs. Each row has $8$ chairs,$1$ behind the other. There are $2$ classes sitting for the examination with $16$ students in each class. It is desired that in each row,all students belong to the same class and that no $2$ adjacent rows are allotted to the same class. In how many ways can these $32$ students be seated?