If $^nC_r$ denotes the number of combinations of $n$ things taken $r$ at a time,then the expression $^nC_{r+1} + ^nC_{r-1} + 2 \times ^nC_r$ equals

$12$ persons are to be arranged at a round table. If two particular persons among them are not to be side by side,the total number of arrangements is

The students in a class are seated according to their marks in the previous examination. Once,it so happens that four of the students got equal marks and therefore the same rank. To decide their seating arrangement,the teacher wants to write down all possible arrangements,one in each of separate bits of paper,in order to choose one of these by lots. How many bits of paper are required?

How many numbers can be formed from the digits $1, 2, 3, 4$ when the repetition is not allowed?

If the four-letter words (need not be meaningful) are to be formed using the letters from the word "$MEDITERRANEAN$" such that the first letter is $R$ and the fourth letter is $E,$ then the total number of all such words is

If seven women and seven men are to be seated around a circular table such that there is a man on either side of every woman,then the number of seating arrangements is