How Many Integral Solutions Are There To The System Of Equations X1 + x2 + x3 + x4 + x5 = 20 And x1 + x2 + x3 = 5 When xk ≥ 0?

Let X is a set containing n elements. A subset P of set X is chosen at random. The set X is then reconstructed by replacing the elements of set Pand another set Q is chosen at random then find the number of ways to form sets such that 

Let X is a set containing n elements. A subset P of set X is chosen at random. The set X is then reconstructed by replacing the elements of set Pand another set Q is chosen at random. Find  number of ways to  chosenP and Q such that P ∪ Q contains exactly r elements.

In how many ways can 12 books be equally distributed among 3 students?

10 different toys are to be distributed among 10 children. Total number of ways of distributing these toys so that exactly 2 children do not get any toy, is equal to:

There are 20 persons among whom are two brothers. Find the number of ways in which we can arrange them around a circle so that there is exactly one person between the two brothers.

In how many ways can a party of 4 men and 4 women be seated at a circular table so that no two women are adjacent?

There are 5 gentlemen and 4 ladies to dine at a round table. In how many ways can they seat themselves so that no two ladies are together?

Find the number of ways in which 8 different flowers can be strung to form a garland so that 4 particular flowers are never separated.

How many integral solutions are there to x + y + z + t = 29, when x ≥ 1, y≥ 2, z ≥ 3 and t ≥ 0?

There are n points in a plane of which no three are in a straight line except ‘m’ which are all in a straight line. Then the number of different quadrilaterals, that can be formed with the given points as vertices, is :