UPSC previous year question paper of subject Permutation, Combination & Probability from the year 2016-2020 with answers.
- #1. How many different sums can be formed with the denominations Rs 50, Rs100, Rs 200, Rs 500 and Rs 2,000 taking at least three denominations at a time? [2020-II]
- #2. How many different 5-letter words (with or without meaning) can be constructed using all the letters of the word ‘DELHI’ so that each word has to start with D and end with I? 
- #3. How many five-digit prime numbers can be obtained by using all the digits 1, 2, 3, 4 and 5 without repetition of digits? 
- #4. One page is torn from a booklet whose pages are numbered in the usual manner starting from the first page as 1. The sum of the numbers on the remaining pages is 195. The torn page contains which of the following numbers? 
- #5. The number of parallelograms that can be formed from a set of four parallel lines intersecting another set of four parallel lines, is 
- #6. Suppose you have sufficient amount of rupee currency in three denominations: Rs 1, Rs 10 and ` 50. In how many different ways can you pay a bill of Rs 107? 
- #7. A printer numbers the pages of a book starting with 1 and uses 3089 digits in all. How many pages does the book have? 
- #8. How many triplets (x, y, z) satisfy the equation x + y + z = 6, where x, y and z are natural numbers? 
- #9. Each face of a cube can be painted in black or white colours. In how many different ways can the cube be painted?
- #10. A bag contains 15 red balls and 20 black balls. Each ball is numbered either 1or 2 or 3. 20% of the red balls are numbered 1 and 40% of them are numbered 3. Similarly, among the black balls, 45% are numbered 2 and 30% are numbered 3. A boy picks a ball at random. He wins if the ball is red and numbered 3 or if it is black and numbered 1 or 2. What are the chances of his winning? 
- #11. While writing all the numbers from 700 to 1000, how many numbers occur in which the digit at hundred’s place is greater than the digit at ten’s place, and the digit at ten’s place is greater than the digit at unit’s place? 
- #12. How many diagonals can be drawn by joining the vertices of an octagon?
- #13. For a sports meet, a winners’ stand comprising three wooden blocks is in the following form :  There are six different colours available to choose from and each of the three wooden blocks is to be painted such that no two of them has the same colour. In how many different ways can the winners’ stand be painted?
- #14. If 2 boys and 2 girls are to be arranged in a row so that the girls are not next to each other, how many possible arrangements are there? 
- #15. A bag contains 20 balls. 8 balls are green, 7 are white and 5 are red. What is the minimum number of balls that must be picked up from the bag blindfolded (without replacing any of it) to be assured of picking at least one ball of each colour? 
- #16. Four-digit numbers are to be formed using the digits 1, 2, 3 and 4; and none of these four digits are repeated in any manner. Further,  1. 2 and 3 are not to immediately follow each other 2. 1 is not to be immediately followed by 3 3. 4 is not to appear at the last place 4. 1 is not to appear at the first place How many different numbers can be formed?