Combinatorics

Question 1

How many onto (or surjective) functions are there from an n-element (n >= 2) set to a 2-element set?

• 2(2ⁿ - 2)
  

• 2ⁿ - 2
  

• 2ⁿ - 1
  

• 2ⁿ
  

Question 2

What is the possible number of reflexive relations on a set of 5 elements?

• 2²⁵
  

• 2²⁰
  

• 2¹⁵
  

• 2¹⁰
  

Question 3

What is the maximum number of different Boolean functions involving n Boolean variables?

• n²

• 2ⁿ

• 2²ⁿ

• 2ⁿ²

Question 4

Suppose that a robot is placed on the Cartesian plane. At each step it is allowed to move either one unit up or one unit right, i.e., if it is at (i,j) then it can move to either (i+1,j) or (i,j+1). 
How many distinct paths are there for the robot to reach the point (10,10) starting from the initial position (0, 0) 

• C
  

• D
  

• B
  

• A
  

Question 5

Consider the data given in above question.
Suppose that the robot is not allowed to traverse the line segment from (4,4) to (5,4). With this constraint, how many distinct paths are there for the robot to reach (10,10) starting from (0,0)? 

• C
  

• D
  

• B
  

• A
  

Question 6

• A
• B
• C
• D

Question 7

Mala has a colouring book in which each English letter is drawn two times. She wants to paint each of these 52 prints with one of k colours, such that the colour-pairs used to colour any two letters are different. Both prints of a letter can also be coloured with the same colour. What is the minimum value of k that satisfies this requirement ?

• 9
  

• 8
  

• 7
  

• 6
  

Question 8

Let A be a sequence of 8 distinct integers sorted in ascending order. How many distinct pairs of sequences, B and C are there such that (i) each is sorted in ascending order, (ii) B has 5 and C has 3 elements, and (iii) the result of merging B and C gives A?

• 256
  

• 56
  

• 30
  

• 2
  

Question 9

n couples are invited to a party with the condition that every husband should be accompanied by his wife. However, a wife need not be accompanied by her husband. The number of different gatherings possible at the party is

• A
• B
• C
• D

Question 10

m identical balls are to be placed in n distinct bags. You are given that m ≥ kn, where, k is a natural number ≥ 1. In how many ways can the balls be placed in the bags if each bag must contain at least k balls? 

• C
  

• D
  

• B
  

• A