GATE Overflow - Recent questions in Combinatory
http://gateoverflow.in/questions/mathematics/discrete-mathematics/combinatory
Powered by Question2AnswerGeneralised permutation combinations
http://gateoverflow.in/126603/generalised-permutation-combinations
In how many ways can a dozen books be placed on four distinguishable shelves<br />
<br />
if no two books are the same, and the positions of the books on the shelves matter?<br />
<br />
(Hint: Break this into 12 tasks, placing each book separately. Start with the sequence 1,2,3,4 to<br />
<br />
represent the shelves. Represent the books by bi, i = 1, 2, ..., 12. Place b1 to the right of one of<br />
<br />
the terms in 1, 2, 3, 4. Then successively place b2, b3, ..., and b12.)Combinatoryhttp://gateoverflow.in/126603/generalised-permutation-combinationsSat, 22 Apr 2017 16:28:10 +0000rosen
http://gateoverflow.in/126428/rosen
<p><big>A shelf holds 12 books in a row. How many ways are there to choose five books so that no two adjacent books are chosen?</big></p>Combinatoryhttp://gateoverflow.in/126428/rosenFri, 21 Apr 2017 10:16:38 +0000Combinatorics
http://gateoverflow.in/126365/combinatorics
A bagel shop has onion bagels, poppy seed bagels, egg bagels, salty bagels, pumpernickel bagels, sesame seed bagels, raisin bagels, and plain bagels. <br />
<br />
How many ways are there to choose<br />
<br />
a dozen bagels with at least three egg bagels and no more than two salty bagels?Combinatoryhttp://gateoverflow.in/126365/combinatoricsThu, 20 Apr 2017 18:56:11 +0000rosen discrete
http://gateoverflow.in/126217/rosen-discrete
How many strings of six lowercase letters of the<br />
English alphabet contain exactly two vowel?Combinatoryhttp://gateoverflow.in/126217/rosen-discreteWed, 19 Apr 2017 12:34:50 +0000Fibonacci series
http://gateoverflow.in/125565/fibonacci-series
Let a,b,c,d are 4 consecutive numbers of Fibonacci series.<br />
<br />
Prove or dis-approve:<br />
<br />
ad-bc= ±1Combinatoryhttp://gateoverflow.in/125565/fibonacci-seriesFri, 14 Apr 2017 16:45:20 +0000Rosen chapter-6 (counting)
http://gateoverflow.in/125539/rosen-chapter-6-counting
How many solutions are there to the equation<br />
x1 + x2 + x3 + x4 + x5 = 21,<br />
where xi , i = 1, 2, 3, 4, 5, is a nonnegative integer such that: 0$\leq$ x1$\leq$10 ?Combinatoryhttp://gateoverflow.in/125539/rosen-chapter-6-countingFri, 14 Apr 2017 11:01:03 +0000How to approach?
http://gateoverflow.in/125197/how-to-approach
A spider is at the bottom of a cliff, and is n inches from the top. Every step it takes brings it one inch closer to the top with probability 1/3, and one inch away from the top with probability 2/3, unless it is at the bottom in which case, it always gets one inch closer. What is the expected number of steps for the spider to reach the top as a function of n?<br />
<br />
a)Never reach to the top<br />
<br />
b)Linear to n<br />
<br />
c)Polynomial to n<br />
<br />
d)Exponential to nCombinatoryhttp://gateoverflow.in/125197/how-to-approachWed, 12 Apr 2017 04:53:36 +0000Rosen, Discrete Mathematics
http://gateoverflow.in/125117/rosen-discrete-mathematics
How many bit strings of length eight contain either three<br />
consecutive 0s or four consecutive 1s?Combinatoryhttp://gateoverflow.in/125117/rosen-discrete-mathematicsTue, 11 Apr 2017 09:45:08 +0000Rosen, Discrete Mathematics ,counting
http://gateoverflow.in/125099/rosen-discrete-mathematics-counting
How many partial functions are there from a set with m elements to a set with n<br />
elements, where m and n are positive integers?Combinatoryhttp://gateoverflow.in/125099/rosen-discrete-mathematics-countingTue, 11 Apr 2017 08:45:26 +0000ISI 2017
http://gateoverflow.in/124372/isi-2017
<p><img alt="" src="http://gateoverflow.in/?qa=blob&qa_blobid=14005501081496295567"></p>Combinatoryhttp://gateoverflow.in/124372/isi-2017Wed, 05 Apr 2017 19:50:21 +0000ISI 2004 MIII
http://gateoverflow.in/123727/isi-2004-miii
Q 4) In how many ways can three person, each throwing a single die once, make a score of 11<br />
<br />
A) 22 B)27 C)24 D)38Combinatoryhttp://gateoverflow.in/123727/isi-2004-miiiMon, 03 Apr 2017 09:27:07 +0000ISI 2004 MIII
http://gateoverflow.in/123721/isi-2004-miii
Q.3 A subset S of set of numbers {2,3,4,5,6,7,8,9,10} is said to be good if has exactly 4 elements and their gcd=1, Then number of good subset is<br />
<br />
A) 126 B) 125 C)123 D)121Combinatoryhttp://gateoverflow.in/123721/isi-2004-miiiMon, 03 Apr 2017 08:57:44 +0000ISI 2004 MIII
http://gateoverflow.in/123718/isi-2004-miii
<p><img alt="" src="http://gateoverflow.in/?qa=blob&qa_blobid=13954498214722884749"></p>Combinatoryhttp://gateoverflow.in/123718/isi-2004-miiiMon, 03 Apr 2017 08:47:03 +0000ISI 2004 MIII
http://gateoverflow.in/123714/isi-2004-miii
Q 1. The number of permutation of {1,2,3,4,5} that keep at least one integer fixed is.<br />
<br />
A) 81 B)76 C)120 D)60Combinatoryhttp://gateoverflow.in/123714/isi-2004-miiiMon, 03 Apr 2017 08:28:33 +0000A bit string is called legitimate if it contains no consecutive zeros, e.g., 0101110 is legitimate,
http://gateoverflow.in/122640/string-legitimate-contains-consecutive-0101110-legitimate
<p>A bit string is called legitimate if it contains no consecutive zeros, e.g., 0101110 is legitimate, whereas 10100111 is not. Let an denote
<br>
the number of legitimate bit strings of length n. Dene a0 = 1. Derive a recurrence relation for an (i.e., express an in terms of the preceding a<sub>i</sub>'s).</p>Combinatoryhttp://gateoverflow.in/122640/string-legitimate-contains-consecutive-0101110-legitimateSun, 26 Mar 2017 18:08:44 +0000Suppose there are n positive real numbers such that their sum is 20
http://gateoverflow.in/122636/suppose-there-are-positive-real-numbers-such-that-their-sum
Suppose there are n positive real numbers such that their sum is 20<br />
and the product is strictly greater than 1. What is the maximum possible<br />
value of n?<br />
<br />
(A) 18 (B) 19 (C) 20 (D) 21Combinatoryhttp://gateoverflow.in/122636/suppose-there-are-positive-real-numbers-such-that-their-sumSun, 26 Mar 2017 17:42:55 +0000The number of terms independent of x in the binomial expansion of
http://gateoverflow.in/122635/the-number-of-terms-independent-of-in-the-binomial-expansion
The number of terms independent of x in the binomial expansion of <br />
<br />
(A) 0 (B) 1 (C) 2 (D) 5Combinatoryhttp://gateoverflow.in/122635/the-number-of-terms-independent-of-in-the-binomial-expansionSun, 26 Mar 2017 17:39:06 +0000Suppose a 6 digit number N is formed by rearranging the digits of the number 123456
http://gateoverflow.in/122634/suppose-digit-number-formed-rearranging-digits-number-123456
Suppose a 6 digit number N is formed by rearranging the digits of the number 123456. If N is divisible by 5, then the set of all possible<br />
remainders when N is divided by 45 is<br />
(A) {30} (B) {15, 30} (C) {0,15,30} (D) {0, 5, 15, 30}Combinatoryhttp://gateoverflow.in/122634/suppose-digit-number-formed-rearranging-digits-number-123456Sun, 26 Mar 2017 17:30:33 +0000A club with n members is organized into four committees so that each member
http://gateoverflow.in/122622/club-with-members-organized-into-four-committees-that-member
A club with n members is organized into four committees so that each<br />
member belongs to exactly two committees and each pair of committees<br />
has exactly one member in common. Then<br />
(A) n = 4<br />
(B) n = 6<br />
(C) n = 8<br />
(D) n cannot be determined from the given informationCombinatoryhttp://gateoverflow.in/122622/club-with-members-organized-into-four-committees-that-memberSun, 26 Mar 2017 15:49:37 +0000What is the highest power of 18 contained
http://gateoverflow.in/122544/what-is-the-highest-power-of-18-contained
What is the highest power of $18$ contained in $50C25$?Combinatoryhttp://gateoverflow.in/122544/what-is-the-highest-power-of-18-containedSat, 25 Mar 2017 13:19:55 +0000MadeEasy Workbook [2016] Q1
http://gateoverflow.in/122518/madeeasy-workbook-2016-q1
In how many ways can seven girls be seated at a round table so that two particular girls are separated?<br />
<br />
(a) 260<br />
<br />
(b) 320<br />
<br />
(c) 480<br />
<br />
(d) 560Combinatoryhttp://gateoverflow.in/122518/madeeasy-workbook-2016-q1Sat, 25 Mar 2017 07:47:51 +0000MadeEasy Workbook[2016] Q48
http://gateoverflow.in/122514/madeeasy-workbook-2016-q48
Find the integer value of x satisfying the inequality$\binom{10}{x-1} < 2\binom{10}{x}$ .Combinatoryhttp://gateoverflow.in/122514/madeeasy-workbook-2016-q48Sat, 25 Mar 2017 07:33:30 +0000MadeEasy Workbook[2016] Q11
http://gateoverflow.in/122513/madeeasy-workbook-2016-q11
In how many different ways can 8 different shirts be distributed among 4 different people so that each receives 2 shirts?<br />
<br />
(a) 2850<br />
<br />
(b) 2680<br />
<br />
(c) 2520<br />
<br />
(d) 1908Combinatoryhttp://gateoverflow.in/122513/madeeasy-workbook-2016-q11Sat, 25 Mar 2017 07:26:46 +0000Recurrence relation and generating function
http://gateoverflow.in/122064/recurrence-relation-and-generating-function
<p>We have two types of shapes.</p>
<p><img alt="" src="http://gateoverflow.in/?qa=blob&qa_blobid=11374212072259770229"></p>
<p>Using these shapes we need to construct $2$*$x$ shapes (height is 2 units and width is $x$ units).</p>
<p>For example, all $5$ possible constructions of $2$*$2$ area are shown below,</p>
<p><img alt="" src="http://gateoverflow.in/?qa=blob&qa_blobid=12417655988685761434"></p>
<p>And following is one possible construction of $2$*$4$ area,</p>
<p><img alt="" src="http://gateoverflow.in/?qa=blob&qa_blobid=3845051216181764483"></p>
<p> </p>
<p>Now if $h_x$ is the no of possible arrangements for an area of $2$*$x$ $\text{unit}^2$ , where $x$ = $0$,$1$,$2$,$3$,$4$.....$\infty$, then find $G(x)$ corresponding to <$h_0$,$h_1$,$h_2$,$h_3$...> </p>Combinatoryhttp://gateoverflow.in/122064/recurrence-relation-and-generating-functionSun, 19 Mar 2017 07:09:21 +0000combinotrics
http://gateoverflow.in/121748/combinotrics
For a set of five true or false question ,no student has written all correct answers, and no two students have given the same given sequence of answers .What is the maximum number of students in the class for this to be possibleCombinatoryhttp://gateoverflow.in/121748/combinotricsThu, 16 Mar 2017 06:09:27 +0000Kenneth Rosen 6.4-44 advanced counting
http://gateoverflow.in/121627/kenneth-rosen-6-4-44-advanced-counting
<ol>
<li>Find out the generating function for $S_n = 1^2 + 2^2 + 3^2 + 4^2 + ... n^2$ and with that generating function show that it is equal to $\begin{align*} \frac{n\left ( n+1 \right )\left ( 2n+1 \right )}{6} \end{align*}$</li>
<li> Find out the generating function for $S_n = 1^3 + 2^3 + 3^3 + 4^3 + ... n^3$ and with that generating function show that it is equal to $\begin{align*} \left ( \frac{n\left ( n+1 \right )}{2} \right )^2 \end{align*}$</li>
</ol>Combinatoryhttp://gateoverflow.in/121627/kenneth-rosen-6-4-44-advanced-countingWed, 15 Mar 2017 04:55:04 +0000Generating function
http://gateoverflow.in/121624/generating-function
Let $h_n$ denote the number of non-negative integral solutions of the equation <br />
<br />
$3x_1 + 4x_2 + 2x_3 + 5x_4 = n$<br />
<br />
Find the generating function $g(x)$ for $h_0,h_1,h_2,h_3 ... h_n$Combinatoryhttp://gateoverflow.in/121624/generating-functionWed, 15 Mar 2017 03:28:53 +0000Discrete Probability Doubt
http://gateoverflow.in/121501/discrete-probability-doubt
Consider a group of k people. Assume that each person's birthday is drawn uniformly at random from the 365 possibilities. (And ignore leap years.) What is the smallest value of ksuch that the expected number of pairs of distinct people with the same birthday is at least one?Combinatoryhttp://gateoverflow.in/121501/discrete-probability-doubtMon, 13 Mar 2017 15:40:18 +0000combinatorics
http://gateoverflow.in/121227/combinatorics
In how many ways can a group of n people be divided into pairs(2 people) ,given that n is an even number ?Combinatoryhttp://gateoverflow.in/121227/combinatoricsFri, 10 Mar 2017 11:26:36 +0000rosen excercise
http://gateoverflow.in/121089/rosen-excercise
How many solutions are there to the equation<br />
x1 + x2 + x3 + x4 + x5 + x6 = 29,<br />
where xi , i = 1, 2, 3, 4, 5, 6, is a nonnegative integer such<br />
that<br />
a) x1 ≤ 5?<br />
b) x1 < 8 and x2 > 8?Combinatoryhttp://gateoverflow.in/121089/rosen-excerciseThu, 09 Mar 2017 10:04:21 +0000How many bit strings of length n contains at least ,at most , exactly r 1's
http://gateoverflow.in/121079/how-many-bit-strings-of-length-contains-at-least-most-exactly
How many bit strings of length n contains 1)at least 2) at most 3) exactly <br />
<br />
r 1'sCombinatoryhttp://gateoverflow.in/121079/how-many-bit-strings-of-length-contains-at-least-most-exactlyThu, 09 Mar 2017 08:17:01 +0000how many solutions are there to equation x1+x2+x3 =11
http://gateoverflow.in/121077/how-many-solutions-are-there-to-equation-x1-x2-x3-11
how many solutions are there to equation<br />
<br />
x1+x2+x3 =11<br />
<br />
with conditions x1<= 1 ,x2<=2 ,x3<=3Combinatoryhttp://gateoverflow.in/121077/how-many-solutions-are-there-to-equation-x1-x2-x3-11Thu, 09 Mar 2017 08:08:37 +0000what is the probability that a randomly chosen bit string of length 10 is palindrome
http://gateoverflow.in/121058/what-probability-randomly-chosen-string-length-palindrome
what is the probability that a randomly chosen bit string of length 10 is palindrome<br />
<br />
a)1/64 b)1/32 c) 1/8 d)1/4Combinatoryhttp://gateoverflow.in/121058/what-probability-randomly-chosen-string-length-palindromeThu, 09 Mar 2017 06:02:11 +0000Rosen excercise
http://gateoverflow.in/121057/rosen-excercise
How many ways are there for 10 women and 6 men to stand in a line so that no two men stand to each otherCombinatoryhttp://gateoverflow.in/121057/rosen-excerciseThu, 09 Mar 2017 05:57:55 +0000k.Rosen excercise. . How many positive integers less than 1000....
http://gateoverflow.in/120950/k-rosen-excercise-how-many-positive-integers-less-than-1000
22. How many positive integers less than 1000<br />
<br />
a) have distinct digits?<br />
b) have distinct digits and are even?Combinatoryhttp://gateoverflow.in/120950/k-rosen-excercise-how-many-positive-integers-less-than-1000Wed, 08 Mar 2017 09:24:39 +0000Manipulation of sum
http://gateoverflow.in/119787/manipulation-of-sum
Prove the identity:<br />
<br />
$$\begin{align*} &\sum_{i=0}^{n}\sum_{j=0}^{i} a_ia_j = \frac{1}{2}\left ( \left ( \sum_{i=0}^{n}a_i \right )^2 + \left ( \sum_{i=0}^{n}a_i^2 \right )\right ) \end{align*}$$Combinatoryhttp://gateoverflow.in/119787/manipulation-of-sumSat, 25 Feb 2017 11:30:07 +0000the number of ways in which 4 distinct balls
http://gateoverflow.in/119753/the-number-of-ways-in-which-4-distinct-balls
the number of ways in which 4 distinct balls can be put in 4 boxes labelled a,b,c,d such that b does not follow a, and c does not follow b, and d does not follow c,isCombinatoryhttp://gateoverflow.in/119753/the-number-of-ways-in-which-4-distinct-ballsFri, 24 Feb 2017 14:47:51 +0000A closet has 5 pair of shoes.
http://gateoverflow.in/119749/a-closet-has-5-pair-of-shoes
A closet has 5 pair of shoes. The number of ways in which 4 shoes can be chosen from it so that there will be no complete pair isCombinatoryhttp://gateoverflow.in/119749/a-closet-has-5-pair-of-shoesFri, 24 Feb 2017 14:39:29 +0000The number of ways of seating three gentlemen
http://gateoverflow.in/119747/the-number-of-ways-of-seating-three-gentlemen
The number of ways of seating three gentlemen and three ladies in a row, such that each gentlemen is adjacent to atleast one lady.Combinatoryhttp://gateoverflow.in/119747/the-number-of-ways-of-seating-three-gentlemenFri, 24 Feb 2017 14:34:25 +0000C. L. Liu 3.38(b)
http://gateoverflow.in/119745/c-l-liu-3-38-b
Among $3n + 1$ objects, $n$ of them are identical. Find the number of ways to select $n$ objects out of these $3n + 1$ objects.Combinatoryhttp://gateoverflow.in/119745/c-l-liu-3-38-bFri, 24 Feb 2017 14:16:00 +0000Let X={a1,a2,...,a7} be a set of seven elements and Y={b1,b2,b3} a set of three elements.
http://gateoverflow.in/119741/let-a2-be-set-of-seven-elements-and-b1-b2-b3-set-three-elements
<p>Let X={a1,a2,...,a7} be a set of seven elements and Y={b1,b2,b3} a set of three elements. The number of functions f from X to Y such that {i} f is onto and {ii}there are exactly three statements x in X such that f(x)=b<sub>1</sub>,is </p>Combinatoryhttp://gateoverflow.in/119741/let-a2-be-set-of-seven-elements-and-b1-b2-b3-set-three-elementsFri, 24 Feb 2017 13:02:47 +0000let S={1,2,...,100}.
http://gateoverflow.in/119740/let-s-1-2-100
let S={1,2,...,100}. The number of nonempty subsets A of S such that the product of elements in A is even isCombinatoryhttp://gateoverflow.in/119740/let-s-1-2-100Fri, 24 Feb 2017 12:52:58 +0000The number of functions f from
http://gateoverflow.in/119739/the-number-of-functions-f-from
The number of functions f from {1,2,...,20} into {1,2,....,20} such that f(k) is a multiple of 3 whenever k is a multiple of 4 isCombinatoryhttp://gateoverflow.in/119739/the-number-of-functions-f-fromFri, 24 Feb 2017 12:42:52 +0000consider the quadratic equation of the form x2+bx+c=0
http://gateoverflow.in/119738/consider-the-quadratic-equation-of-the-form-x2-bx-c-0
consider the quadratic equation of the form x2+bx+c=0.The number of such equations that have real roots and coefficients b and c from the set{1,2,3,4,5} (b and c may be equal) isCombinatoryhttp://gateoverflow.in/119738/consider-the-quadratic-equation-of-the-form-x2-bx-c-0Fri, 24 Feb 2017 12:36:13 +0000lets A1,A2,A3 be three points on a straight line
http://gateoverflow.in/119737/lets-a1-a2-a3-be-three-points-on-a-straight-line
lets A1,A2,A3 be three points on a straight line. Lets B1,B2,B3,B4,B5 be five points on a straight line parallel to first one. Each of the three points on the first line is joined by a straight line to each of the five points on the second line. Further,no three or more of these joining lines met at a point except possibly at the A's or B's. Then the number of point of intersection of the joining lines lying between the two given straight lineCombinatoryhttp://gateoverflow.in/119737/lets-a1-a2-a3-be-three-points-on-a-straight-lineFri, 24 Feb 2017 12:20:34 +0000Maths
http://gateoverflow.in/119721/maths
There are 11 points on a plane with 5 lying on one straight line and another 5 lying on other straight line which is parallel to the first line. The remaining point is not collinear with any two of the previous points.The number of triangles that can be formed with vertices chosen from these 11 points isCombinatoryhttp://gateoverflow.in/119721/mathsFri, 24 Feb 2017 08:32:29 +0000In a multiple-choice test there are 6 questions.
http://gateoverflow.in/119676/in-a-multiple-choice-test-there-are-6-questions
<p>In a multiple-choice test there are 6 questions. 4 alternatives answers are given for each question by choosing one answer for each question, then the number of ways to get exactly 4 correct answers is
<br>
(A) <img alt="4^6-4^2" height="15" src="http://s0.wp.com/latex.php?zoom=1.5&latex=4%5E6-4%5E2&bg=ffffff&fg=000&s=0" width="49">;
<br>
(B) 135;
<br>
(C) 9;
<br>
(D) 120.</p>Combinatoryhttp://gateoverflow.in/119676/in-a-multiple-choice-test-there-are-6-questionsThu, 23 Feb 2017 03:39:18 +0000There are 7 identical white balls and 3 identical black balls.
http://gateoverflow.in/119675/there-are-7-identical-white-balls-and-identical-black-balls
<p>There are 7 identical white balls and 3 identical black balls. The number of distinguishable arrangements in a row of all the balls, so that no two black balls are adjacent, is
<br>
(A) 120;
<br>
(B) 89(8!);
<br>
(C) 56;
<br>
(D) 42x<img alt="5^4" height="14" src="http://s0.wp.com/latex.php?zoom=1.5&latex=5%5E4&bg=ffffff&fg=000&s=0" width="14">.</p>Combinatoryhttp://gateoverflow.in/119675/there-are-7-identical-white-balls-and-identical-black-ballsThu, 23 Feb 2017 03:38:12 +0000In an examination, the score in each of the four languages
http://gateoverflow.in/119674/in-an-examination-the-score-in-each-of-the-four-languages
In an examination, the score in each of the four languages – Bengali, Hindi, Urdu and Telegu- can be integers between 0 and 10. Then the number of ways in which a student can secure a total score of 21 is<br />
(A) 880;<br />
(B) 760;<br />
(C) 450;<br />
(D) 1360.Combinatoryhttp://gateoverflow.in/119674/in-an-examination-the-score-in-each-of-the-four-languagesThu, 23 Feb 2017 03:34:43 +0000The number of ways in which three non-negative integers
http://gateoverflow.in/119673/the-number-of-ways-in-which-three-non-negative-integers
<p>The number of ways in which three non-negative integers <img alt="n_1,n_2,n_3" height="11" src="http://s0.wp.com/latex.php?zoom=1.5&latex=n_1%2Cn_2%2Cn_3&bg=ffffff&fg=000&s=0" width="63"> can be chosen such that <img alt="n_1+n_2+n_3=10" height="16" src="http://s0.wp.com/latex.php?zoom=1.5&latex=n_1%2Bn_2%2Bn_3%3D10&bg=ffffff&fg=000&s=0" width="127"> is</p>Combinatoryhttp://gateoverflow.in/119673/the-number-of-ways-in-which-three-non-negative-integersThu, 23 Feb 2017 03:32:21 +0000