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. [Hint: Required number of arrangement =⁷C₄× 5!]

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 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.[Hint: Required number = 210 – 1].

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.

[Hint: Required number of ways =³C₁×⁶C₂+³C₂x ⁶C₁+³C₃.]