Permutations an...

At an election, a voter may vote for any number of candidates not greater than the number to be chosen. There are 10 candidates and 5 members are to be chosen. The number of ways in which a voter may vote for at least one candidate is given by

The number of positive integers with the property that they can be expressed as the sum of the cubes of 2 positive integers in two different ways is

A class has 21 students. The class teacher has been asked to make n groups of r students each and go to zoo taking one group at a time. The size of group i.e. the value of r for which the teacher goes to the zoo, maximum number of times is (no group can go to the zoo twice)

There are p, q, r points on three parallel lines L₁, L₂ and L₃ all of which lie in one plane. The number of triangles which can be formed with vertices at these points is

The number of solutions if x + y + z + w = 20, if x ≥ 1, y ≥ 2, z ≥ 3, w ≥ 4, is

The number of ten balls selections, if at least one ball of each colour must be included in the 10 ball selections is

n books are arranged on a shelf so that two particular books are not next to each other, There were 480 arrangements altogether. Then the number of books on the shelf is ________