GATE Overflow - Recent questions tagged counting
https://gateoverflow.in/tag/counting
Powered by Question2AnswerP and C
https://gateoverflow.in/161929/p-and-c
Right triangle PQR is to be constructed in the xy-plane so that the right angle is at P<br />
and line PR is parallel to the x-axis. The x and y coordinates of P, Q and R are to be<br />
integers that satisfy the inequalities :– $-4\leq x \leq 5$ and $6\leq y \leq 16$ . How many different triangles could be<br />
constructed with these properties ?<br />
(a) 110 (b) 1,100 (c) 9,900 (d) 10,000Mathematical Logichttps://gateoverflow.in/161929/p-and-cSun, 22 Oct 2017 06:28:51 +0000K. Rosen: Countings
https://gateoverflow.in/159361/k-rosen-countings
<p><strong>Question</strong> How many different strings of length 2 can be made from the letters in '<strong>ORONO</strong>'?</p>
<p>I am getting 7 strings as follows: OO, ON, OR, NO, RO, RN, NR
<br>
but given<strong> answer is 6</strong>.</p>
<p> </p>Combinatoryhttps://gateoverflow.in/159361/k-rosen-countingsThu, 12 Oct 2017 05:16:46 +0000counting
https://gateoverflow.in/140204/counting
<p>Let G be a complete undirected graph on 6 vertices. If vertices of G are labeled, then the number of distinct cycles of length 4 in G is equal to</p>
<ol>
<li>15</li>
<li>30</li>
<li>90</li>
<li>360</li>
</ol>Graph Theoryhttps://gateoverflow.in/140204/countingThu, 27 Jul 2017 11:00:45 +0000gatebook
https://gateoverflow.in/136871/gatebook
How many bit strings of length $6$ have more no of zeros than ones??<br />
<br />
We can solve this drawing tree but it takes so much time Is there any other way??Combinatoryhttps://gateoverflow.in/136871/gatebookSun, 09 Jul 2017 08:47:05 +0000Counting
https://gateoverflow.in/132945/counting
How many ways are there to put six temporary employees into four labeled offices so that there is at least one temporary employee in each of these four offices?Mathematical Logichttps://gateoverflow.in/132945/countingWed, 14 Jun 2017 00:21:55 +0000ISBN9780070681880 - Exercise 5.3 - Problem 35 (Kenneth Rosen 7ed SIE)
https://gateoverflow.in/131070/isbn9780070681880-exercise-problem-kenneth-rosen-7ed-sie
How many bit strings contain exactly eight 0s and 10 1s if every 0 must be immediately followed by a 1 ?Mathematical Logichttps://gateoverflow.in/131070/isbn9780070681880-exercise-problem-kenneth-rosen-7ed-sieSat, 27 May 2017 10:20:41 +0000counting
https://gateoverflow.in/130233/counting
Number of words of 4 letters that can be formed with the letters of the word IITJEE is<br />
<br />
a) 42 b) 82 c)102 d) 142Probabilityhttps://gateoverflow.in/130233/countingSat, 20 May 2017 12:04:28 +0000#rosen , #counting , #5.1 , 41(a)
https://gateoverflow.in/128608/%23rosen-%23counting-%235-1-41-a
in how many ways can a photographer at a wedding arrange 6 people in a row from a group of 10 people, where the bride and the groom are among these 10 people , if<br />
<br />
a) the bride must next to the groom<br />
<br />
my solution is since given that bride and groom must be next to each other then now we have to select 4 person from 8 that is 8c4 now we have to arrange these people since bride and groom next to each other then take BG as one block and rest four __ , __ , __ , __ ,<br />
<br />
now arranging BG , __ , __ ,__, __ will take 5! , and arranging BG mutually is 2! ,<br />
<br />
so total ways is 8c4*5!*2!<br />
<br />
but answer is given 240 (rosen )Combinatoryhttps://gateoverflow.in/128608/%23rosen-%23counting-%235-1-41-aSun, 07 May 2017 17:51:30 +0000how to remove ambiguity that whether should take lower or upper shield in counting problem
https://gateoverflow.in/128445/remove-ambiguity-whether-should-lower-shield-counting-problem
how many positive integer between 50 and 100 ,<br />
<br />
(a) divisible by 7<br />
(b) divisible by 11<br />
(c) divisible by 7 and 11 ?<br />
<br />
people generally answer this question as<br />
<br />
(a) lowershield [((100-50)-1 )/7] = 7 , yess its true the no is 56 , 63 , 70, 84, 91 , 98 <br />
(b) lowershield [((100-50)-1 )/11] = 4 , oops its not true the no is 55,66,77,88,99<br />
<br />
(c) lowershield [((100-50)-1 )/lcm(7,11)] = 0 , oops its not true the no is 77<br />
<br />
if u will say to take uppershield then lets come<br />
<br />
(a) uppershield[((100-50)-1)/7]=7 , yess its true<br />
<br />
(b) uppershield[((100-50)-1)/11]=5 , yess its true<br />
<br />
(c) uppershield[((100-50)-1)/lcm(7,11)]=1 , yess its true<br />
<br />
now come to another question<br />
<br />
how many no is divisible between 5 to 31 is divisible by 4 ?<br />
<br />
case1 : loweshield [((31-5)-1)/4]=6 , yess its true 8,12,16,20,24,28<br />
<br />
case 2 : uppershierld [((31-5)-1/4)]=7 , oops its wrong<br />
<br />
how can we remove this ambiguity ?<br />
<br />
if u will argue that check the no ? but we have to check for large no like 50 to 300000 then how can u count ?<br />
<br />
my doubt is , is any way to remove this ambiguity ?Mathematical Logichttps://gateoverflow.in/128445/remove-ambiguity-whether-should-lower-shield-counting-problemSun, 07 May 2017 09:54:44 +0000No. of DFA's Possible
https://gateoverflow.in/115952/no-of-dfas-possible
The number of different DFA's with two states X and Y,where X is the initial state,over the alphabet $\sum$ = {0,1,2}Theory of Computationhttps://gateoverflow.in/115952/no-of-dfas-possibleSat, 04 Feb 2017 23:07:34 +0000Set Counting [GateBook]
https://gateoverflow.in/110031/set-counting-gatebook
<p><img alt="" src="http://gateoverflow.in/?qa=blob&qa_blobid=11626069763951897866"></p>Set Theory & Algebrahttps://gateoverflow.in/110031/set-counting-gatebookTue, 24 Jan 2017 06:54:04 +0000answer 90 or 444?
https://gateoverflow.in/106281/answer-90-or-444
<p><img alt="" src="http://gateoverflow.in/?qa=blob&qa_blobid=6347837754231132418"></p>Combinatoryhttps://gateoverflow.in/106281/answer-90-or-444Mon, 16 Jan 2017 13:55:15 +0000Counting possible no of subsets from a set of numbers S
https://gateoverflow.in/102944/counting-possible-no-of-subsets-from-a-set-of-numbers-s
<ul>
<li>Using numbers from <strong>S</strong> = $\left \{ 1,2,3,4.......n \right \}$</li>
<li>We can use maximum up to <span class="marker">m</span> numbers to form a <span class="marker">set</span> using numbers from <strong>S. </strong>Repetition of numbers allowed.</li>
<li>How many ways we can form a set such that, $\sum x_i = K$. Where $K$ is another positive integer. Where $x_i$ are the elements belong $S$ that are included in the newly formed <span class="marker">set.</span></li>
</ul>
<p><strong>For example :</strong></p>
<ul>
<li>S = $\left \{ 1,2,3,4,5...11,12 \right \}$</li>
<li><span class="marker"> m</span> = $4$</li>
<li>if $K = 6$</li>
<li>Then possible few possible sets are $\{2,4\}, \;\; \{1,3,2\}, \;\; \{1,4,1\},\;\; \{1,1,1,3\}$ etc.</li>
<li>$\{1,1,1,1,2\}$ is not valid <span class="marker">set</span> for example.</li>
<li>Now how many such <span class="marker">sets</span> for a particular instance of the problem ? with </li>
<li>S = $\left \{ 1,2,3,4,5,6...12 \right \}$ , $m = 5$, $K = 8$ ?</li>
<li>If there is any generic idea ?</li>
<li>Ordered / Unordered both the cases !</li>
</ul>Combinatoryhttps://gateoverflow.in/102944/counting-possible-no-of-subsets-from-a-set-of-numbers-sMon, 09 Jan 2017 11:11:15 +0000ace-test-series
https://gateoverflow.in/101887/ace-test-series
<p><img alt="" src="http://gateoverflow.in/?qa=blob&qa_blobid=10593045386689658062"></p>
<p>According to my understanding, there should be men and women both in the team. So we can do:
<br>
3M and 1W or
<br>
2M and 2W or
<br>
1M and 3W.</p>
<p>So it will be: C(5,3)*C(5,1)+C(5,2)*C(5,2)+C(5,1)*C(5,3).</p>
<p>But the answer given is 600. How is it possible?</p>Probabilityhttps://gateoverflow.in/101887/ace-test-seriesSat, 07 Jan 2017 10:38:07 +0000test series counting
https://gateoverflow.in/100202/test-series-counting
The number of pairs of set (X, Y) are there that satisfy the condition X, Y ⊆ {1, 2, 3,<br />
4, 5, 6} and X ∩ Y = Φ ________.Combinatoryhttps://gateoverflow.in/100202/test-series-countingWed, 04 Jan 2017 02:12:43 +0000Made easy #counting
https://gateoverflow.in/100189/made-easy-%23counting
Consider a bitstring of length 10 containing only 0 and 1. The number of string<br />
contain 3 consicutive 0’s or 3 consicutive 1’s are ________Numerical Abilityhttps://gateoverflow.in/100189/made-easy-%23countingWed, 04 Jan 2017 01:13:56 +0000madeeasy
https://gateoverflow.in/97579/madeeasy
An entrepreneur needs to assign 5 different tasks to three of his employees. If every employee is assigned at least 1 task, how many ways can the entrepreneur assign those tasks to his employees?<br />
<br />
Doubt: Can this question reduce to distributing labelled objects into labelled boxes or distributing labelled objects into unlabelled boxes with no boxes are empty.<br />
<br />
My view: The three employees are different from each other and hence should be treated as labelled boxes.Combinatoryhttps://gateoverflow.in/97579/madeeasyWed, 28 Dec 2016 08:35:03 +0000TIFR2017-B-12
https://gateoverflow.in/95819/tifr2017-b-12
<p>An undirected graph is complete if there is an edge between every pair of vertices. Given a complete undirected graph on $n$ vertices, in how many ways can you choose a direction for the edges so that there are no directed cycles?</p>
<ol style="list-style-type: upper-alpha;">
<li>$n$</li>
<li>$\frac{n(n-1)}{2}$</li>
<li>$n!$</li>
<li>$2^n$</li>
<li>$2^m, \: \text{ where } m=\frac{n(n-1)}{2}$</li>
</ol>Graph Theoryhttps://gateoverflow.in/95819/tifr2017-b-12Fri, 23 Dec 2016 17:28:27 +0000Kenneth Rosen (Special Indian Edition) Section 5.1 Exercise Problem # 5
https://gateoverflow.in/94477/kenneth-rosen-special-indian-edition-section-exercise-problem
<p><strong>Question</strong>: Six different airlines fly from New York to Denver and seven fly from Denver to San Francisco. How many different pairs of airlines can you choose on which to book a trip from New York to San Francisco via Denver, when you pick an airline for the flight to Denver and an airline for the continuation flight to San Francisco ? How many of these pairs involve more than one airline ?</p>
<p> </p>Combinatoryhttps://gateoverflow.in/94477/kenneth-rosen-special-indian-edition-section-exercise-problemMon, 19 Dec 2016 19:53:26 +0000Counting
https://gateoverflow.in/94304/counting
How many ways, can sum be equal to 12 of 3 dice?<br />
<br />
Solution:<br />
x1+x2+x3=12 <br />
where 1<=x1<=6;1<=x2<=6; 1<=x3<=6<br />
How to solve it further?Combinatoryhttps://gateoverflow.in/94304/countingMon, 19 Dec 2016 11:28:36 +0000Consider the graph G whose vertices are 4 element subsets of the set {1, 2, 3…10}
https://gateoverflow.in/87546/consider-the-graph-whose-vertices-are-element-subsets-the-set
Consider the graph G whose vertices are 4 element subsets of the set {1, 2, 3…10} with two vertices adjacent if and only if their intersection is empty. Then the number of edges does G have _______Graph Theoryhttps://gateoverflow.in/87546/consider-the-graph-whose-vertices-are-element-subsets-the-setMon, 28 Nov 2016 22:53:15 +0000A arrives at office at 8-10am regularly; B arrives at 9-11 am every day
https://gateoverflow.in/85624/arrives-at-office-at-10am-regularly-arrives-at-11-am-every-day
A arrives at office at 8-10am regularly; B arrives at 9-11 am every day. Probability that one day B arrives before A? [Assume arrival time of both A and B are uniformly distributed]Probabilityhttps://gateoverflow.in/85624/arrives-at-office-at-10am-regularly-arrives-at-11-am-every-dayThu, 24 Nov 2016 19:13:09 +00002..You are in a game show! There are 10 closed doors, 0 leads to nothing and 1 leads to an expensive sports car.
https://gateoverflow.in/85603/game-there-closed-doors-leads-nothing-leads-expensive-sports
<p>You are in a game show! There are 10 closed doors, 0 leads to nothing and 1 leads to an expensive sports car. You are allowed to pick a door and earn the sports car if it’s behind the door you choose. You choose a door and the host tells you he was preauthorized to make your chance of winning better! You have two options.
<br>
Option 1: Get the right to open two doors and win if the car is behind either of the ones you open.
<br>
Option 2: Have the host open 4 empty doors [None of them the one you had choose] and then get the right to switch if you want.
<br>
If you want to win the car, what should you do?</p>
<ol>
<li> You should be indifferent</li>
<li> Go with option 2, then don’t switch</li>
<li> Go with option 2 and switch</li>
<li> Go with option 1</li>
</ol>Probabilityhttps://gateoverflow.in/85603/game-there-closed-doors-leads-nothing-leads-expensive-sportsThu, 24 Nov 2016 17:05:18 +0000There are 10 bacteria in a flask. Every hour 3 bacteria die and the remaining ones are each divided into 2 after 1 day
https://gateoverflow.in/79841/there-bacteria-flask-every-bacteria-remaining-divided-after
<p>There are 10 bacteria in a flask. Every hour 3 bacteria die and the remaining ones are each divided into 2 after 1 day, how many bacteria will live there?
<br>
Assume that the flask is large enough to contain any number of bacteria?</p>
<ol>
<li> 2<sup>26</sup></li>
<li> 2<sup>26</sup> + 6</li>
<li> 2<sup>24</sup> + 6</li>
<li> 2<sup>24</sup></li>
</ol>Combinatoryhttps://gateoverflow.in/79841/there-bacteria-flask-every-bacteria-remaining-divided-afterMon, 07 Nov 2016 16:12:36 +0000A palindrome is a string whose reversal is identical to the string. How many bit strings of length n are palindromes?
https://gateoverflow.in/79824/palindrome-reversal-identical-strings-length-palindromes
<p>A palindrome is a string whose reversal is identical to the string. How many bit strings of length n are palindromes?</p>
<ol>
<li> 2<sup>⌈n⁄2⌉</sup></li>
<li> 2<sup>(⌊ n/2⌋ )</sup></li>
<li> 2<sup>⌈n⁄2⌉</sup> -1</li>
<li> 2<sup>(⌊ n/2⌋)</sup> -1</li>
</ol>Combinatoryhttps://gateoverflow.in/79824/palindrome-reversal-identical-strings-length-palindromesMon, 07 Nov 2016 14:25:29 +0000Expectated no of coin toss
https://gateoverflow.in/69392/expectated-no-of-coin-toss
An unbiased coin is tossed repeatedly and outcomes are recorded. What is the expected no of toss to get HT ( one head and one tail consecutively) ?Probabilityhttps://gateoverflow.in/69392/expectated-no-of-coin-tossSat, 24 Sep 2016 06:52:49 +0000generalised pigeonhole principle
https://gateoverflow.in/64718/generalised-pigeonhole-principle
Show that if seven integers are selected from the first<br />
10 positive integers, there must be at least two pairs<br />
of these integers with the sum 11.<br />
<br />
Attempt-:partition will be {(1,10),(2,9),(3,8)(4,7)(5,6)}<br />
<br />
now how to apply pigeonhole principle to find the answer?Combinatoryhttps://gateoverflow.in/64718/generalised-pigeonhole-principleThu, 25 Aug 2016 00:36:08 +0000Kenneth Rosen
https://gateoverflow.in/64381/kenneth-rosen
Suppose that each pair of a genetically engineered species of rabbits left on an island produces two new pairs of rabbits at the age of 1 month and six new pairs of rabbits at the age of 2 months and every month afterward. None of the rabbits ever die or leave the island. <br />
<br />
Find a recurrence relation for the number of pairs of rabbits on the island n months after one new born pair is left on the island.https://gateoverflow.in/64381/kenneth-rosenTue, 23 Aug 2016 06:02:23 +0000CMI2013-B-02
https://gateoverflow.in/46612/cmi2013-b-02
A complete graph on $n$ vertices is an undirected graph in which every pair of distinct vertices is connected by an edge. A simple path in a graph is one in which no vertex is repeated. Let $G$ be a complete graph on 10 vertices. Let $u, \: v, \: w$ be three distinct vertices in $G$. How many simple paths are there from $u$ to $v$ going through $w$?Graph Theoryhttps://gateoverflow.in/46612/cmi2013-b-02Mon, 23 May 2016 14:35:31 +0000Gate2015 Aptitude Set 8 Q8
https://gateoverflow.in/40183/gate2015-aptitude-set-8-q8
How many four digit numbers can be formed with the 10 digits $0, 1, 2, \ldots, 9$ if no number can start with 0 and if repetitions are not allowed?Numerical Abilityhttps://gateoverflow.in/40183/gate2015-aptitude-set-8-q8Mon, 15 Feb 2016 14:44:39 +0000In how many ways can 2n+1 seats in a congress be divided among 3 parties ?
https://gateoverflow.in/38063/in-how-many-ways-can-2n-seats-congress-divided-among-parties
In how many ways can $2n+1$ seats in a congress be divided among 3 parties so that coalition of any 2 parties will ensure them majority?Combinatoryhttps://gateoverflow.in/38063/in-how-many-ways-can-2n-seats-congress-divided-among-partiesWed, 27 Jan 2016 12:23:41 +0000Let L = {a, bb} How many strings of length 10 are present in L* ?
https://gateoverflow.in/15026/let-l-a-bb-how-many-strings-of-length-10-are-present-in-l
Let L = {a, bb}<br />
How many strings of length 10 are present in L* ?Theory of Computationhttps://gateoverflow.in/15026/let-l-a-bb-how-many-strings-of-length-10-are-present-in-lSat, 15 Aug 2015 16:20:25 +0000permutaions and combinations
https://gateoverflow.in/13257/permutaions-and-combinations
<p><span style="font-size:16px">How many bit strings contain exactly eight 0s and 10 1s if every 0 must be immediately followed by a 1?</span></p>
<p><span style="font-size:16px">I got answer 9C2=36.Answer given 45</span></p>
Combinatoryhttps://gateoverflow.in/13257/permutaions-and-combinationsWed, 15 Jul 2015 18:42:27 +0000pigeonhole
https://gateoverflow.in/13186/pigeonhole
<p><span style="font-size:16px">Prove that at a party where there are at least two people, there are two people who know the same number of other people there.</span></p>
Combinatoryhttps://gateoverflow.in/13186/pigeonholeTue, 14 Jul 2015 11:13:49 +0000pigeonhole
https://gateoverflow.in/13185/pigeonhole
<p><span style="font-size:16px">Show that there are at least six people in California (population: 37 million) with the same three initials who were born on the same day of the year (but not necessarily in the same year). Assume that everyone has three initials.</span></p>
Combinatoryhttps://gateoverflow.in/13185/pigeonholeTue, 14 Jul 2015 11:09:21 +0000pigeonhole
https://gateoverflow.in/13184/pigeonhole
Show that in a group of 10 people (where any two people are either friends or enemies), there are either three mutual friends or four mutual enemies, and there are either three mutual enemies or four mutual friends.Combinatoryhttps://gateoverflow.in/13184/pigeonholeTue, 14 Jul 2015 11:07:15 +0000pigeonhole
https://gateoverflow.in/13183/pigeonhole
<p><span style="font-size:16px">Show that in a group of five people (where any two people are either friends or enemies), there are not necessarily three mutual friends or three mutual enemies.</span></p>
Combinatoryhttps://gateoverflow.in/13183/pigeonholeTue, 14 Jul 2015 11:04:50 +0000pigeonhole
https://gateoverflow.in/13180/pigeonhole
<p><span style="font-size:16px">Assume that in a group of six people, each pair of individuals consists of two friends or two enemies. Show that there are either three mutual friends or three mutual enemies in the group.</span></p>
Combinatoryhttps://gateoverflow.in/13180/pigeonholeTue, 14 Jul 2015 10:49:55 +0000application of pigeonhole principle
https://gateoverflow.in/13170/application-of-pigeonhole-principle
<p><span style="font-size:16px">During a month with 30 days, a baseball team plays at least one game a day, but no more <span style="line-height:1.6">than 45 games. Show that there must be a period of some number of consecutive days during </span><span style="line-height:1.6">which the team must play exactly 14 games</span></span></p>
Combinatoryhttps://gateoverflow.in/13170/application-of-pigeonhole-principleTue, 14 Jul 2015 07:47:01 +0000counting
https://gateoverflow.in/13129/counting
How many bit strings of length 10 contain either five consecutive 0s or five consecutive 1s?<br />
<br />
I got 382.Is it correct?Combinatoryhttps://gateoverflow.in/13129/countingMon, 13 Jul 2015 10:48:44 +0000number of function
https://gateoverflow.in/13128/number-of-function
<p><span style="font-size:16px">How many functions are there from the set {1, 2, . . . , n}, where n is a positive integer, to the set {0, 1}
<br>
a) that assign 1 to exactly one of the positive integers less than n?</span></p>
Combinatoryhttps://gateoverflow.in/13128/number-of-functionMon, 13 Jul 2015 10:39:52 +0000counting
https://gateoverflow.in/13127/counting
How many bit strings of length eight contain either three consecutive 0s or four consecutive 1s?Combinatoryhttps://gateoverflow.in/13127/countingMon, 13 Jul 2015 10:26:31 +0000Please explain this question .. Number of trivial substrings in “GATE2013” are: A. 37 B. 35 C. 2 D. 36
https://gateoverflow.in/11136/please-explain-question-number-trivial-substrings-gate2013
Combinatoryhttps://gateoverflow.in/11136/please-explain-question-number-trivial-substrings-gate2013Sun, 07 Jun 2015 10:53:44 +0000How many ways are there to choose a dozen donuts from 20 varieties
https://gateoverflow.in/4579/how-many-ways-are-there-to-choose-dozen-donuts-from-varieties
<p>How many ways are there to choose a dozen donuts from 20 varieties</p>
<ol style="list-style-type: lower-alpha;">
<li>if all donuts are of the same variety?</li>
<li>if there are at least two varieties among the dozen donuts chosen? </li>
<li><span style="line-height: 1.6;">if there must be at least six blueberry-filled donuts? </span></li>
<li><span style="line-height: 1.6;">if there can be no more than six blueberry-filled donuts?</span></li>
</ol>
Combinatoryhttps://gateoverflow.in/4579/how-many-ways-are-there-to-choose-dozen-donuts-from-varietiesMon, 24 Nov 2014 13:40:04 +0000GATE1994_1.6, ISRO2008-29
https://gateoverflow.in/2443/gate1994_1-6-isro2008-29
<p>The number of distinct simple graphs with up to three nodes is</p>
<ol style="list-style-type: upper-alpha;">
<li>15</li>
<li>10</li>
<li>7</li>
<li>9</li>
</ol>Graph Theoryhttps://gateoverflow.in/2443/gate1994_1-6-isro2008-29Sat, 04 Oct 2014 11:18:26 +0000GATE2005-35
https://gateoverflow.in/1371/gate2005-35
<p>How many distinct binary search trees can be created out of 4 distinct keys?</p>
<ol style="list-style-type:upper-alpha">
<li>5</li>
<li>14</li>
<li>24</li>
<li>42</li>
</ol>Graph Theoryhttps://gateoverflow.in/1371/gate2005-35Mon, 22 Sep 2014 23:36:28 +0000GATE2005-44
https://gateoverflow.in/1170/gate2005-44
<p>What is the minimum number of ordered pairs of non-negative numbers that should be chosen to ensure that there are two pairs (a,b) and (c,d) in the chosen set such that $$a \equiv c \mod \;3 \;and \;b \equiv d \mod \;5$$</p>
<ol style="list-style-type:upper-alpha">
<li>4</li>
<li>6</li>
<li>16</li>
<li>24</li>
</ol>Set Theory & Algebrahttps://gateoverflow.in/1170/gate2005-44Sun, 21 Sep 2014 11:48:52 +0000GATE2004-79
https://gateoverflow.in/1073/gate2004-79
<p>How many graphs on $n$ labeled vertices exist which have at least $\frac{(n^2 - 3n)}{ 2}$ edges ?</p>
<ol style="list-style-type: upper-alpha;">
<li>${}^{\left(\frac{n^2-n}{2}\right)}C_{\frac{n^2-3n} {2}}$</li>
<li>${\sum_{k=0}^{\left (\frac{n^2-3n}{2} \right )}} .^{\left(n^2-n\right)}C_k$</li>
<li>${}^{\left(\frac{n^2-n}{2}\right)}C_n$</li>
<li>${\sum_{k=0}^n}.^{\left(\frac{n^2-n}{2}\right)}C_k$</li>
</ol>Graph Theoryhttps://gateoverflow.in/1073/gate2004-79Fri, 19 Sep 2014 01:36:46 +0000GATE2001-2.15
https://gateoverflow.in/733/gate2001-2-15
<p>How many undirected graphs (not necessarily connected) can be constructed out of a given set $V=\{v_1, v_2, \dots v_n\}$ of $n$ vertices?</p>
<ol style="list-style-type: upper-alpha;">
<li>$\frac{n(n-1)} {2}$</li>
<li>$2^n$</li>
<li>$n!$</li>
<li>$2^\frac{n(n-1)} {2} $</li>
</ol>Graph Theoryhttps://gateoverflow.in/733/gate2001-2-15Mon, 15 Sep 2014 00:04:19 +0000GATE2012-38
https://gateoverflow.in/473/gate2012-38
<p>Let G be a complete undirected graph on 6 vertices. If vertices of G are labeled, then the number of distinct cycles of length 4 in G is equal to</p>
<ol style="list-style-type: upper-alpha;">
<li>15</li>
<li>30</li>
<li>90</li>
<li>360</li>
</ol>Graph Theoryhttps://gateoverflow.in/473/gate2012-38Fri, 12 Sep 2014 17:54:10 +0000