Combinatorics - GeeksforGeeks

21 Jan, 2014
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 
Total possible number of functions is 2n. In mathematics, a function f from a set X to a set Y is surjective (or onto), or a surjection, if every element y in Y has a corresponding element x in X such that f(x) = yThere are total 2 functions out of 2n that are NOT onto: one that maps to all 1s and other that maps to all 2s. Therefore total number of onto functions is 2n - 2.  
What is the possible number of reflexive relations on a set of 5 elements? 
225 
220 
215 
210 
Number of reflexive relations is 2n2-n which is 220 for n = 5 
No of inputs sequences possible for a n variable Boolean function = 2n

Each input sequence can give either T or F as output ( 2 possible values )

So, Total no of Boolean functions are -

2X2X2X2X2X2X.............X2X2X2X2X2X2

<-------------------- 2n Times -------------->

22n

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)  
 
D 
C 
B 
A 
At each move, robot can move either 1 unit right, or 1 unit up, and there will be 20 such moves required to reach (10,10) from (0,0). So we have to divide these 20 moves, numbered from 1 to 20, into 2 groups: right group and up group. Right group contains those moves in which we move right, and up group contains those moves in which we move up. Each group contains 10 elements each. So basically, we have to divide 20 things into 2 groups of 10 10 things each, i.e., we need to find all possible arrangements of {r, r, r, r, r, r, r, r, r, r, u, u, u, u, u, u, u, u, u, u} where r represents right move and u represents up move. The arrangements can can be done in 20! / (10!∗10!) = 20C10 ways. So option (A) is correct.  
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)?  
 
D 
C 
B 
A 
Since we are not allowed to traverse from (4,4) to (5,4), we subtract all those paths which were passing through (4,4) to (5,4). To count number of paths passing through (4,4) to (5,4), we find number of paths from (0,0) to (4,4), and then from (5,4) to (10,10).  
From (0,0) to (4,4), number of paths = 8C4
[found in same way as in previous question].

From (5,4) to (10,10), number of paths = 11C5.
So total number of paths required : 20C10 − 8C4 ∗ 11C5.
So option (D) is correct. 
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 
This question is slightly ambiguous. So first let us understand what question is asking. So in a book, we have letters A-Z and each letter is printed twice, so there are 52 letters. Now we have to color each letter, so we need a pair of colors for that, because each letter is printed twice. Also in a pair, both colors can be some. Now condition is that a pair of colors can't be used more than once. So suppose Mala has 3 colors : Red, Blue, Green. She can color as follows : (A,A) : (Red,Red), (B,B) : (Blue,Blue), (C,C) : (Green,Green), (D,D) : (Red,Blue), (E,E) : (Red,Green), (F,F) : (Blue,Green). Now we don't have more pairs of colors left, we have used all pairs, but could color only 6 letters out of 26. So question is to find minimum no. of colors, so that we could color all 26 letters. So if Mala has k colors, she can have k pairs of same colors, thus coloring k letters, then kC2 other pairs of colors, thus coloring kC2 more letters. So total no. of letters colored = k+kC2=k+k(k−1)2=k(k+1)2. So we want k(k+1)2≥26 i.e. k(k+1)≥52, so k≥7, so option (C) is correct. 
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 
Suppose you have selected 3 elements from 8 in 8C3 ways, the remaining elements are treated as another array and merging both the arrays gives the sorted array. Here, you can select either 3 or 5. => 8C3 = 8C5 = 8!/(3!5!) = 7*8 = 56 Ways.  
1) Nobody goes to gathering
2) Wife alone goes
2) Both go
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?  
 
D 
C 
B 
A 
This is very simple application of stars and bars. Since we want atleast k balls in each bag, so first we put kn balls into bags, k balls in each bag. Now we are left with m - kn balls, and we have to put them into n bags such that each bag may receive 0 or more balls. So applying theorem 2 of stars and bars with m - nk stars and n bars, we get number of ways to be m−kn+n-1 Cn−1. So option (B) is correct. 
Writing code in comment? Please use ide.geeksforgeeks.org, generate link and share the link here.
Python Program for Breadth First Search or BFS for a Graph
Best Time to Buy and Sell Stock
Must Do Coding Questions for Product Based Companies
How to calculate MOVING AVERAGE in a Pandas DataFrame?
What is "network ID" and "host ID" in IP Addresses?
What is Transmission Control Protocol (TCP)?
How to Calculate Number of Host in a Subnet?
Python OpenCV - Canny() Function
How to Convert Categorical Variable to Numeric in Pandas?
How to Replace Values in Column Based on Condition in Pandas?