Find the number of permutations of n distinct things taken r together, in which 3 particular things must occur together.

 

Find the number of different words that can be formed from the letters of the word ‘TRIANGLE’ so that no vowels are together.

Find the number of positive integers greater than 6000 and less than 7000 which are divisible by 5, provided that no digit is to be repeated.

There are 10 persons named P1
, P2 , P3 , ... P10. Out of 10 persons, 5 persons are to be arranged in a line such that in each arrangement P1
 must occur whereas P4 and P5
 do not occur. Find the number of such possible arrangements.

There are 10 lamps in a hall. Each one of them can be switched on independently. Find the number of ways in which the hall can be illuminated.

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

Find the number of integers greater than 7000 that can be formed with the digits 3, 5, 7, 8 and 9 where no digits are repeated.

If 20 lines are drawn in a plane such that no two of them are parallel and no three are concurrent, in how many points will they intersect each other?

In a certain city, all telephone numbers have six digits, the first two digits always being 41 or 42 or 46 or 62 or 64. How many telephone numbers have all six digits
distinct?

In an examination, a student has to answer 4 questions out of 5 questions; questions 1 and 2 are however compulsory. Determine the number of ways in which the
student can make the choice.

A convex polygon has 44 diagonals. Find the number of its sides