GATE Overflow - Recent questions tagged counting
http://gateoverflow.in/tag/counting
Powered by Question2AnswerNo. of DFA's Possible
http://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 Computationhttp://gateoverflow.in/115952/no-of-dfas-possibleSat, 04 Feb 2017 17:37:34 +0000Set Counting [GateBook]
http://gateoverflow.in/110031/set-counting-gatebook
<p><img alt="" src="http://gateoverflow.in/?qa=blob&qa_blobid=11626069763951897866"></p>Set Theory & Algebrahttp://gateoverflow.in/110031/set-counting-gatebookTue, 24 Jan 2017 01:24:04 +0000answer 90 or 444?
http://gateoverflow.in/106281/answer-90-or-444
<p><img alt="" src="http://gateoverflow.in/?qa=blob&qa_blobid=6347837754231132418"></p>Combinatoryhttp://gateoverflow.in/106281/answer-90-or-444Mon, 16 Jan 2017 08:25:15 +0000Counting possible no of subsets from a set of numbers S
http://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>Combinatoryhttp://gateoverflow.in/102944/counting-possible-no-of-subsets-from-a-set-of-numbers-sMon, 09 Jan 2017 05:41:15 +0000ace-test-series
http://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>Probabilityhttp://gateoverflow.in/101887/ace-test-seriesSat, 07 Jan 2017 05:08:07 +0000test series counting
http://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 ________.Combinatoryhttp://gateoverflow.in/100202/test-series-countingTue, 03 Jan 2017 20:42:43 +0000Made easy #counting
http://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 Abilityhttp://gateoverflow.in/100189/made-easy-%23countingTue, 03 Jan 2017 19:43:56 +0000madeeasy
http://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.Combinatoryhttp://gateoverflow.in/97579/madeeasyWed, 28 Dec 2016 03:05:03 +0000Kenneth Rosen (Special Indian Edition) Section 5.1 Exercise Problem # 5
http://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>Combinatoryhttp://gateoverflow.in/94477/kenneth-rosen-special-indian-edition-section-exercise-problemMon, 19 Dec 2016 14:23:26 +0000Counting
http://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?Combinatoryhttp://gateoverflow.in/94304/countingMon, 19 Dec 2016 05:58:36 +0000Consider the graph G whose vertices are 4 element subsets of the set {1, 2, 3…10}
http://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 Theoryhttp://gateoverflow.in/87546/consider-the-graph-whose-vertices-are-element-subsets-the-setMon, 28 Nov 2016 17:23:15 +0000GATE1989-1-v
http://gateoverflow.in/87051/gate1989-1-v
The number of possible commutative binary operations that can be defined on a set of $n$ elements (for a given n) is ___________.Combinatoryhttp://gateoverflow.in/87051/gate1989-1-vSun, 27 Nov 2016 10:52:41 +0000A arrives at office at 8-10am regularly; B arrives at 9-11 am every day
http://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]Probabilityhttp://gateoverflow.in/85624/arrives-at-office-at-10am-regularly-arrives-at-11-am-every-dayThu, 24 Nov 2016 13:43:09 +00002..You are in a game show! There are 10 closed doors, 0 leads to nothing and 1 leads to an expensive sports car.
http://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>Probabilityhttp://gateoverflow.in/85603/game-there-closed-doors-leads-nothing-leads-expensive-sportsThu, 24 Nov 2016 11:35:18 +0000There are 10 bacteria in a flask. Every hour 3 bacteria die and the remaining ones are each divided into 2 after 1 day
http://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>Combinatoryhttp://gateoverflow.in/79841/there-bacteria-flask-every-bacteria-remaining-divided-afterMon, 07 Nov 2016 10:42:36 +0000A palindrome is a string whose reversal is identical to the string. How many bit strings of length n are palindromes?
http://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>Combinatoryhttp://gateoverflow.in/79824/palindrome-reversal-identical-strings-length-palindromesMon, 07 Nov 2016 08:55:29 +0000Expectated no of coin toss
http://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) ?Probabilityhttp://gateoverflow.in/69392/expectated-no-of-coin-tossSat, 24 Sep 2016 01:22:49 +0000generalised pigeonhole principle
http://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?Combinatoryhttp://gateoverflow.in/64718/generalised-pigeonhole-principleWed, 24 Aug 2016 19:06:08 +0000Kenneth Rosen
http://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.http://gateoverflow.in/64381/kenneth-rosenTue, 23 Aug 2016 00:32:23 +0000Gate 2015 Aptitude Set 8 Q8
http://gateoverflow.in/40183/gate-2015-aptitude-set-8-q8
<p> </p>
<p><span style="font-family: ;font-size:10pt;color:rgb(0,0,0);font-style:normal;font-variant:normal;">Q.8 How many four digit numbers can be formed with the 10 digits 0, 1, 2, ..., 9 if no number can start
<br><span style="font-family: ;font-size:10pt;color:rgb(0,0,0);font-style:normal;font-variant:normal;">with 0 and if repetitions are not allowed?</span></span>
<br style="font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: -webkit-auto; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px;">
</p>
Numerical Abilityhttp://gateoverflow.in/40183/gate-2015-aptitude-set-8-q8Mon, 15 Feb 2016 09:14:39 +0000In how many ways can 2n+1 seats in a congress be divided among 3 parties ?
http://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?Combinatoryhttp://gateoverflow.in/38063/in-how-many-ways-can-2n-seats-congress-divided-among-partiesWed, 27 Jan 2016 06:53:41 +0000TIFR 2015 A-13
http://gateoverflow.in/29069/tifr-2015-a-13
<p><img alt="" src="http://gateoverflow.in/?qa=blob&qa_blobid=467344726710769866"></p>
Combinatoryhttp://gateoverflow.in/29069/tifr-2015-a-13Tue, 01 Dec 2015 23:45:09 +0000Let L = {a, bb} How many strings of length 10 are present in L* ?
http://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 Computationhttp://gateoverflow.in/15026/let-l-a-bb-how-many-strings-of-length-10-are-present-in-lSat, 15 Aug 2015 10:50:25 +0000permutaions and combinations
http://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>
Combinatoryhttp://gateoverflow.in/13257/permutaions-and-combinationsWed, 15 Jul 2015 13:12:27 +0000pigeonhole
http://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>
Combinatoryhttp://gateoverflow.in/13186/pigeonholeTue, 14 Jul 2015 05:43:49 +0000pigeonhole
http://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>
Combinatoryhttp://gateoverflow.in/13185/pigeonholeTue, 14 Jul 2015 05:39:21 +0000pigeonhole
http://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.Combinatoryhttp://gateoverflow.in/13184/pigeonholeTue, 14 Jul 2015 05:37:15 +0000pigeonhole
http://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>
Combinatoryhttp://gateoverflow.in/13183/pigeonholeTue, 14 Jul 2015 05:34:50 +0000pigeonhole
http://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>
Combinatoryhttp://gateoverflow.in/13180/pigeonholeTue, 14 Jul 2015 05:19:55 +0000application of pigeonhole principle
http://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>
Combinatoryhttp://gateoverflow.in/13170/application-of-pigeonhole-principleTue, 14 Jul 2015 02:17:01 +0000counting
http://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?Combinatoryhttp://gateoverflow.in/13129/countingMon, 13 Jul 2015 05:18:44 +0000number of function
http://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>
Combinatoryhttp://gateoverflow.in/13128/number-of-functionMon, 13 Jul 2015 05:09:52 +0000counting
http://gateoverflow.in/13127/counting
How many bit strings of length eight contain either three consecutive 0s or four consecutive 1s?Combinatoryhttp://gateoverflow.in/13127/countingMon, 13 Jul 2015 04:56:31 +0000Please explain this question .. Number of trivial substrings in “GATE2013” are: A. 37 B. 35 C. 2 D. 36
http://gateoverflow.in/11136/please-explain-question-number-trivial-substrings-gate2013
Combinatoryhttp://gateoverflow.in/11136/please-explain-question-number-trivial-substrings-gate2013Sun, 07 Jun 2015 05:23:44 +0000GATE2015-2_40
http://gateoverflow.in/8212/gate2015-2_40
The number of onto functions (surjective functions) from set $X = \{1, 2, 3, 4\}$ to set $Y=\{a,b,c\}$ is ______.Set Theory & Algebrahttp://gateoverflow.in/8212/gate2015-2_40Thu, 12 Feb 2015 23:25:54 +0000GATE2015-2_9
http://gateoverflow.in/8058/gate2015-2_9
The number of divisors of 2100 is ____.Combinatoryhttp://gateoverflow.in/8058/gate2015-2_9Thu, 12 Feb 2015 06:03:55 +0000How many ways are there to choose a dozen donuts from 20 varieties
http://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>
Combinatoryhttp://gateoverflow.in/4579/how-many-ways-are-there-to-choose-dozen-donuts-from-varietiesMon, 24 Nov 2014 08:10:04 +0000GATE2006-IT-24
http://gateoverflow.in/3563/gate2006-it-24
<p>What is the cardinality of the set of integers X defined below?</p>
<p>X = {n | 1 ≤ n ≤ 123, n is not divisible by either 2, 3 or 5}</p>
<ol style="list-style-type: upper-alpha;">
<li>28</li>
<li>33</li>
<li>37</li>
<li>44</li>
</ol>Set Theory & Algebrahttp://gateoverflow.in/3563/gate2006-it-24Fri, 31 Oct 2014 03:28:47 +0000GATE1995_7
http://gateoverflow.in/2642/gate1995_7
<ol style="list-style-type: lower-alpha;">
<li>
<p>Determine the number of divisors of 600.</p>
</li>
<li>
<p>Compute without using power series expansion $\lim_{x \to 0} \frac{\sin x}{x}$</p>
</li>
</ol>
<p> </p>
Calculushttp://gateoverflow.in/2642/gate1995_7Wed, 08 Oct 2014 18:10:57 +0000GATE2005-44
http://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 & Algebrahttp://gateoverflow.in/1170/gate2005-44Sun, 21 Sep 2014 06:18:52 +0000GATE2000-5
http://gateoverflow.in/676/gate2000-5
<p>A multiset is an unordered collection of elements where elements may repeat any number of times. The size of a multiset is the number of elements in it, counting repetitions.</p>
<ol style="list-style-type:lower-alpha">
<li>What is the number of multisets of size 4 that can be constructed from n distinct elements so that at least one element occurs exactly twice?</li>
<li>How many multisets can be constructed from n distinct elements? </li>
</ol>DShttp://gateoverflow.in/676/gate2000-5Sun, 14 Sep 2014 16:58:38 +0000GATE2000-3
http://gateoverflow.in/674/gate2000-3
Consider the following sequence:<br />
<br />
$s_1 = s_2 = 1$ and $s_i = 1 + \min \left({s_{i-1}, s_{i-2}}\right) \text{ for } i > 2$.<br />
<br />
Prove by induction on $n$ that $s_n=⌈\frac{n}{2}⌉$.Combinatoryhttp://gateoverflow.in/674/gate2000-3Sun, 14 Sep 2014 16:57:23 +0000