GATE Overflow - Recent questions tagged combinatory
http://gateoverflow.in/tag/combinatory
Powered by Question2AnswerGenerating Function
http://gateoverflow.in/104105/generating-function
<p>Find the coefficient of <strong>x<sup>83</sup></strong> in<strong> (x<sup>5</sup>+ x<sup>8</sup>+ x<sup>11</sup>+ x<sup>14</sup>+ x<sup>17</sup>)<sup>10 </sup></strong>?</p>Combinatoryhttp://gateoverflow.in/104105/generating-functionWed, 11 Jan 2017 14:16:05 +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 +0000Stars and bar problem
http://gateoverflow.in/100948/stars-and-bar-problem
<p>Can someone explain stars and bar problme using suitable example and images
<br>
<br>
Problem : <a rel="nofollow" href="https://en.wikipedia.org/wiki/Stars_and_bars_%28combinatorics%29#Proofs_via_the_method_of_stars_and_bars">https://en.wikipedia.org/wiki/Stars_and_bars_%28combinatorics%29#Proofs_via_the_method_of_stars_and_bars</a></p>
<p> </p>Combinatoryhttp://gateoverflow.in/100948/stars-and-bar-problemThu, 05 Jan 2017 07:34:19 +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 +0000Number of ways ... distinguishable into distinguishable.
http://gateoverflow.in/99437/number-of-ways-distinguishable-into-distinguishable
How many ways can n books be placed on k distinguishable shelves if no two books are same and the position of the books on the shelves matter.Mathematical Logichttp://gateoverflow.in/99437/number-of-ways-distinguishable-into-distinguishableMon, 02 Jan 2017 08:15:22 +0000Counting
http://gateoverflow.in/99333/counting
How many eight digit numbers are there, that contain a 5 and a 6____________? please explain!<br />
<br />
Ans: 8486912Combinatoryhttp://gateoverflow.in/99333/countingMon, 02 Jan 2017 05:01:11 +0000TheTrevTutor discrete maths 2 videos
http://gateoverflow.in/98563/thetrevtutor-discrete-maths-2-videos
<p>I want to know how good are the videos of discrete maths by TheTrevTutor. Has anyone been following the videos while preparing for GATE? </p>
<p> </p>
<p><a rel="nofollow" href="https://www.youtube.com/watch?v=DBugSTeX1zw&list=PLDDGPdw7e6Aj0amDsYInT_8p6xTSTGEi2">https://www.youtube.com/watch?v=DBugSTeX1zw&list=PLDDGPdw7e6Aj0amDsYInT_8p6xTSTGEi2</a></p>Combinatoryhttp://gateoverflow.in/98563/thetrevtutor-discrete-maths-2-videosFri, 30 Dec 2016 19:11:02 +0000combinations
http://gateoverflow.in/97711/combinations
<p><img alt="" src="http://gateoverflow.in/?qa=blob&qa_blobid=11787487673859343352"></p>Mathematical Logichttp://gateoverflow.in/97711/combinationsWed, 28 Dec 2016 10:58:23 +0000TIFR2016-A-15
http://gateoverflow.in/97624/tifr2016-a-15
<p>In a tournament with 7 teams, each team plays one match with every other team. For each match, the team earns two points if it wins, one point if it ties, and no points if it loses. At the end of all matches, the teams are ordered in the descending order of their total points (the order among the teams with the same total are determined by a whimsical tournament referee). The first three teams in this ordering are then chosen to play in the next round. What is the minimum total number of points a team must earn in order to be guaranteed a place in the next round?</p>
<ol style="list-style-type:upper-alpha">
<li>13</li>
<li>12</li>
<li>11</li>
<li>10</li>
<li>9</li>
</ol>Combinatoryhttp://gateoverflow.in/97624/tifr2016-a-15Wed, 28 Dec 2016 07:03:13 +0000Kenneth Rosen (Special Indian Edition) Section 6.1 Exercise Problem #9d
http://gateoverflow.in/95581/kenneth-rosen-special-indian-edition-section-exercise-problem
Solve the recurrence relation $a_n = a_{n-1} + 2n + 3, a_0 = 4$Combinatoryhttp://gateoverflow.in/95581/kenneth-rosen-special-indian-edition-section-exercise-problemThu, 22 Dec 2016 17:22:54 +0000TIFR2017-A-6
http://gateoverflow.in/95033/tifr2017-a-6
<p>How many disctict words can be formed by permuting the letters of the word ABRACADABRA?</p>
<ol style="list-style-type:upper-alpha">
<li>$\frac{11!}{5! \: 2! \: 2!}$</li>
<li>$\frac{11!}{5! \: 4! }$</li>
<li>$11! \: 5! \: 2! \: 2!\:$</li>
<li>$11! \: 5! \: 4!$</li>
<li>$11! $</li>
</ol>Combinatoryhttp://gateoverflow.in/95033/tifr2017-a-6Wed, 21 Dec 2016 11:52:18 +0000TIFR2017-A-5
http://gateoverflow.in/94953/tifr2017-a-5
<p>How many distinct ways are there to split 50 identical coins among three people so that each person gets at least 5 coins?</p>
<ol style="list-style-type:upper-alpha">
<li>$3^{35}$</li>
<li>$3^{50}-2^{50}$</li>
<li>$\begin{pmatrix} 35 \\ 2 \end{pmatrix}$</li>
<li>$\begin{pmatrix} 50 \\ 15 \end{pmatrix}. 3^{35}$</li>
<li>$\begin{pmatrix} 37 \\ 2 \end{pmatrix}$</li>
</ol>Combinatoryhttp://gateoverflow.in/94953/tifr2017-a-5Wed, 21 Dec 2016 07:07:16 +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/94382/counting
How many ways you can put 3 identical balls into 2 box so that each box has at-most 2 balls?<br />
<br />
Is the answer 2?Combinatoryhttp://gateoverflow.in/94382/countingMon, 19 Dec 2016 08:18:25 +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 +0000binomial theorem and expansions
http://gateoverflow.in/91878/binomial-theorem-and-expansions
<p>can anyone please explain these things:</p>
<p>formula for (1-x)<sup>n</sup></p>
<p>formula for 1/(1-x)<sup>n</sup></p>
<p>general term in expansion of (1-x)<sup>n </sup>and 1/(1-x)<sup>n</sup></p>
<p>and coeffecient of a term in these expansions.</p>
<p>please elaborate a little because i have read few questions on generating functions and binomial where these things are used but i am getting very confused.i dun know much about them and gathering info from internet is also confusing me.</p>Combinatoryhttp://gateoverflow.in/91878/binomial-theorem-and-expansionsSun, 11 Dec 2016 20:23:48 +0000Permutation and combination
http://gateoverflow.in/90501/permutation-and-combination
The number of ways can 10 balls be selected from urn contain 10 identical red balls 5 identical green balls and 3 identical blue balls ?Combinatoryhttp://gateoverflow.in/90501/permutation-and-combinationWed, 07 Dec 2016 14:37:41 +0000Discrete Maths
http://gateoverflow.in/89753/discrete-maths
Explain each one of the following:<br />
<br />
a ) In how many ways can we put 31 people in 3 rooms such that each room has an odd number of people ? <br />
<br />
b ) Coefficient of $x^4$ in the expansion $(1+ x + x^2 + x^3)^{11}$ using generating functions.<br />
<br />
c)Find out number of solutions $x_1+x_2+x_3 = 20 , 2<x_1<6 , 6<x_2<10 , 0<x_3<5$Mathematical Logichttp://gateoverflow.in/89753/discrete-mathsMon, 05 Dec 2016 16:34:04 +0000Permutation and combinations
http://gateoverflow.in/89167/permutation-and-combinations
Number of binary strings of length 10 with 3 consecutive 0's or 1's is ?Combinatoryhttp://gateoverflow.in/89167/permutation-and-combinationsSat, 03 Dec 2016 19:33:12 +00002 bulbs out of a sample of 10 bulbs manufactured by a company are defective.
http://gateoverflow.in/88215/bulbs-out-sample-bulbs-manufactured-company-are-defective
2 bulbs out of a sample of 10 bulbs manufactured by a company are defective. The probability that 3 out of 4 bulbs bought by a customer will not be defective isProbabilityhttp://gateoverflow.in/88215/bulbs-out-sample-bulbs-manufactured-company-are-defectiveWed, 30 Nov 2016 18:08:13 +0000GATE1989-4-i
http://gateoverflow.in/87874/gate1989-4-i
Provide short answers to the following questions:<br />
<br />
How many substrings (of all lengths inclusive) can be formed from a character string of length $n$? Assume all characters to be distinct, prove your answer.Combinatoryhttp://gateoverflow.in/87874/gate1989-4-iTue, 29 Nov 2016 18:52:00 +0000If any 6 points are chosen on the perimeter of a circle,
http://gateoverflow.in/87558/if-any-6-points-are-chosen-on-the-perimeter-of-a-circle
The largest integer n that satisfies the following condition:<br />
If any 6 points are chosen on the perimeter of a circle, then we can draw semicircle of the circle, such that there are at least n points on it. ____Combinatoryhttp://gateoverflow.in/87558/if-any-6-points-are-chosen-on-the-perimeter-of-a-circleMon, 28 Nov 2016 18:31:11 +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 +0000Expected valu
http://gateoverflow.in/86239/expected-valu
<p><strong>I always do expected value questions wrong. Please suggest a detailed way to solve these questions. </strong></p>
<p>Suppose a coin is tossed 9 times, with the result</p>
<p>$HHHT T T T HT$</p>
<p>The first set of three heads is called a run.There are three more runs in this sequence, namely the next four tails, the next head, and the next tail. We do not consider the first two tosses to constitute a run, since the third toss has the same value as the first two.</p>
<p>Now suppose an experiment consists of tossing a fair coin three times. Find the expected number of runs</p>Combinatoryhttp://gateoverflow.in/86239/expected-valuSat, 26 Nov 2016 03:36:33 +0000Generalized permutation and combination
http://gateoverflow.in/85604/generalized-permutation-and-combination
How many positive integers less than 1,000,000 have the sum of their digits equal to 19? (using generating function)Combinatoryhttp://gateoverflow.in/85604/generalized-permutation-and-combinationThu, 24 Nov 2016 11:36:17 +0000GATE1990-3-ix
http://gateoverflow.in/84841/gate1990-3-ix
<p>Choose the correct alternatives (More than one may be correct).</p>
<p>The number of ways in which 5 A's, 5 B's and 5 C's can be arranged in a row is:</p>
<ol style="list-style-type:upper-alpha">
<li>$15!/(5!)^{3}$</li>
<li>$15!$</li>
<li>$\left(\frac{15}{5}\right)$</li>
<li>$15!(5!3!)$.</li>
</ol>Combinatoryhttp://gateoverflow.in/84841/gate1990-3-ixTue, 22 Nov 2016 13:55:36 +0000probability
http://gateoverflow.in/83453/probability
the letters of the word PROBABILITY are arranged in all possible ways . the chance that B's and also two I's occur together is .?Combinatoryhttp://gateoverflow.in/83453/probabilityThu, 17 Nov 2016 15:46:14 +0000how many ways can Harry select 24 of these candies so that he has at least six black one
http://gateoverflow.in/81166/many-ways-harry-select-these-candies-that-has-least-six-black
<p>If there is an unlimited number of red, green, blue and black jelly beans in how many ways can Harry select 24 of these candies so that he has at least six black one? _________.(</p>
<ol>
<li> <img alt="image:EM4/O11a.PNG" height="53" src="http://gatexcel.co.in/cache/EM4/O11a.PNG" width="172"></li>
<li> <img alt="image:EM4/O11b.PNG" height="57" src="http://gatexcel.co.in/cache/EM4/O11b.PNG" width="179"> </li>
<li> <img alt="image:EM4/O11d.PNG" height="49" src="http://gatexcel.co.in/cache/EM4/O11d.PNG" width="183"></li>
</ol>
<p> 4 <img alt="image:EM4/O11c.PNG" src="http://gatexcel.co.in/cache/EM4/O11c.PNG"></p>Combinatoryhttp://gateoverflow.in/81166/many-ways-harry-select-these-candies-that-has-least-six-blackFri, 11 Nov 2016 04:43:58 +0000Discrete Maths
http://gateoverflow.in/81145/discrete-maths
Messages are transmitted through a communication channel using two signals. The transmission of one signal requires $1 $ microsecond, the transmission of other signal requires $2$ microseconds. Assume that each signal in a message is immediately followed by the next signal. The number of different messages consisting of sequences of these two signals that can be sent in $10 \mu s$ is _________ ?<br />
<br />
Please explain ...?Combinatoryhttp://gateoverflow.in/81145/discrete-mathsFri, 11 Nov 2016 02:32:47 +0000GATE1987-1-II
http://gateoverflow.in/80032/gate1987-1-ii
<p>The total number of Boolean functions which can be realised with four variables is:</p>
<ol style="list-style-type:upper-alpha">
<li>$4$</li>
<li>$17$</li>
<li>$256$</li>
<li>$65, 536$</li>
</ol>Digital Logichttp://gateoverflow.in/80032/gate1987-1-iiMon, 07 Nov 2016 18:18:52 +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 +0000Combinatory
http://gateoverflow.in/79840/combinatory
In a country, there are coins of denominations ${$2}$, ${$3}$ and ${$7}$. How many different ways are there to pay exactly ${$10}$ ?Combinatoryhttp://gateoverflow.in/79840/combinatoryMon, 07 Nov 2016 10:12:40 +0000Permutation of coloring sides of a pentagon
http://gateoverflow.in/76864/permutation-of-coloring-sides-of-a-pentagon
Number of ways of painting a regular pentagon with 5 different colors is<br />
<br />
(A) 5!<br />
(B) 12<br />
(C) 1<br />
(D) 4!CO & Architecturehttp://gateoverflow.in/76864/permutation-of-coloring-sides-of-a-pentagonWed, 26 Oct 2016 05:35:05 +0000Mathematics
http://gateoverflow.in/75832/mathematics
How many derangements of (1 2 3 4 5 6 ) end with integer 1 2 and 3 in some order ?Combinatoryhttp://gateoverflow.in/75832/mathematicsFri, 21 Oct 2016 20:48:46 +0000logic functions
http://gateoverflow.in/75390/logic-functions
<p>Given $x,y,z$ are Boolean variables and $f(x,y,z)=f(y',x',z)$. How many such functions are possible with $x, y, z$?</p>
<ol style="list-style-type: upper-alpha;">
<li>$0$</li>
<li>$2^2$</li>
<li>$2^4$</li>
<li>$2^6$ </li>
</ol>Set Theory & Algebrahttp://gateoverflow.in/75390/logic-functionsThu, 20 Oct 2016 01:42:05 +0000counting
http://gateoverflow.in/72351/counting
How many integers less than 1000<br />
<br />
a) are div by 7?<br />
<br />
b) are div by 7 but not by 11<br />
<br />
c) are div by 7 and 11 ?<br />
<br />
d) are div either by 7 or 11 ?<br />
<br />
e) are div exactly by 7 or 11 ?<br />
<br />
f ) are div neither by 7 nor by 11<br />
<br />
g) have distinct digits <br />
<br />
h) have distinct digits and are evenCombinatoryhttp://gateoverflow.in/72351/countingFri, 07 Oct 2016 13:33:22 +0000Expected Value #sheldon and ross
http://gateoverflow.in/70126/expected-value-%23sheldon-and-ross
If n balls are randomly selected from an urn containing N balls of which m are white,find the expected number of white balls selected.Probabilityhttp://gateoverflow.in/70126/expected-value-%23sheldon-and-rossTue, 27 Sep 2016 08:48:49 +0000Counting: Non negative integer solution
http://gateoverflow.in/68939/counting-non-negative-integer-solution
Number of non negative integer solution for<br />
1. x1+x2+x3+x4+x5<10<br />
2.x1+x2+x3+x4+x5=<10<br />
3. x1+x2+x3+x4+x5=10Combinatoryhttp://gateoverflow.in/68939/counting-non-negative-integer-solutionWed, 21 Sep 2016 10:36:14 +0000Discreet Math
http://gateoverflow.in/67961/discreet-math
Number of 10 bit binary string where the string contains 3 consecutive 0's or 3 consecutive 1's?Combinatoryhttp://gateoverflow.in/67961/discreet-mathFri, 16 Sep 2016 08:51:01 +0000Permutation and combination
http://gateoverflow.in/66090/permutation-and-combination
Find the no. of seven digit integers with sum of digits = 11 and formed by using the digits 1,2 and 3 only ?Combinatoryhttp://gateoverflow.in/66090/permutation-and-combinationFri, 02 Sep 2016 16:24:13 +0000Find the number of integral solutions using generating function
http://gateoverflow.in/65803/find-number-integral-solutions-using-generating-function
Find the number of integral soultions of $2x + y + z = 20$ with $x, y, z >= 0$ ?<br />
<br />
I tried finding coefficient of $x^{20}$ in $(1 + x^2 + x^3 + ...... + x^{10})(1+ x^2 + .......... + x^{20})^2$, but it gives wrong answer ?Combinatoryhttp://gateoverflow.in/65803/find-number-integral-solutions-using-generating-functionThu, 01 Sep 2016 00:21:24 +0000#Deragement How many ways we put letter
http://gateoverflow.in/65605/%23deragement-how-many-ways-we-put-letter
<h3><strong>How many ways we can put 5 letters L<sub>1</sub> ,L2, L<sub>3,</sub> L<sub>4 </sub>,L<sub>5 </sub>in 5 envelope e1, e2, e3, e4, e5 (at 1 letter per envelope) so that </strong></h3>
<ol>
<li><strong>At most 1 letter is correctly placed</strong></li>
<li><strong>At least 1 Letter is wrongly placed</strong></li>
</ol>Mathematical Logichttp://gateoverflow.in/65605/%23deragement-how-many-ways-we-put-letterTue, 30 Aug 2016 17:42:43 +0000Permutation
http://gateoverflow.in/65385/permutation
Given a integer N greater than zero.<br />
<br />
How many sequences of 1's and 2's are there such that sum of the numbers in the sequence = N ?<br />
<br />
(not necessary that every sequence must contain both 1 and 2 )<br />
<br />
example :<br />
<br />
for N = 2 ; 11,2 => ans = 2 sequences of 1's and 2's<br />
<br />
for N = 3 ; 11,12,21 => ans = 3 sequences of 1's and 2'sCombinatoryhttp://gateoverflow.in/65385/permutationMon, 29 Aug 2016 15:28:44 +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 +0000Counting
http://gateoverflow.in/64544/counting
How many ways are there to seat six people around a circular table where two seatings are considered the same<br />
when everyone has the same two neighbors without regard to whether they are right or left neighbors?<br />
It would be better if one provide a pictorial explanation.Combinatoryhttp://gateoverflow.in/64544/countingTue, 23 Aug 2016 17:36:51 +0000Counting Theory
http://gateoverflow.in/64288/counting-theory
<p>10 Points were selected in the sequence with each side 1 unit, there will be atleast two points (of those points) such that distance between them can not exceeds..</p>
<ol>
<li>1/(2.sqrt(2))</li>
<li>(sqrt(2))/3</li>
<li>1/3</li>
<li>(sqrt(2))/9</li>
</ol>
<p> </p>Combinatoryhttp://gateoverflow.in/64288/counting-theoryMon, 22 Aug 2016 14:17:00 +0000Counting Theory
http://gateoverflow.in/64286/counting-theory
<p>5 points were selected in an equilateral triangle with each side 1 units. There will at least 2 points (of those points) such that the distance between them can not exceed </p>
<ol>
<li>1/2</li>
<li>1/3</li>
<li>1/4</li>
<li>None</li>
</ol>Combinatoryhttp://gateoverflow.in/64286/counting-theoryMon, 22 Aug 2016 14:12:57 +0000UGCNET-June2016-III-31
http://gateoverflow.in/63952/ugcnet-june2016-iii-31
<p>The number of different binary trees with 6 nodes is</p>
<ol style="list-style-type:upper-alpha">
<li>6</li>
<li>42</li>
<li>132</li>
<li>256</li>
</ol>Combinatoryhttp://gateoverflow.in/63952/ugcnet-june2016-iii-31Sat, 20 Aug 2016 08:23:37 +0000