GATE Overflow - Recent questions in Combinatory
http://gateoverflow.in/questions/mathematics/discrete-mathematics/combinatory
Powered by Question2AnswerRecurrence 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 +0000Generating function
http://gateoverflow.in/119622/generating-function
<p>The coefficient of <img alt="x^4" height="14" src="http://s0.wp.com/latex.php?zoom=1.5&latex=x%5E4&bg=ffffff&fg=000&s=0" width="15"> in the expansion of <img alt="(1+x-2x^2)^7" height="18" src="http://s0.wp.com/latex.php?zoom=1.5&latex=%281%2Bx-2x%5E2%29%5E7&bg=ffffff&fg=000&s=0" width="99"> is</p>Combinatoryhttp://gateoverflow.in/119622/generating-functionWed, 22 Feb 2017 05:02:22 +0000#combinatorics #aptitude
http://gateoverflow.in/118663/%23combinatorics-%23aptitude
<p><img alt="" src="http://gateoverflow.in/?qa=blob&qa_blobid=7688279075571954362"></p>Combinatoryhttp://gateoverflow.in/118663/%23combinatorics-%23aptitudeTue, 14 Feb 2017 10:49:34 +0000GATE2017-2-47
http://gateoverflow.in/118392/gate2017-2-47
If the ordinary generating function of a sequence $\big \{a_n\big \}_{n=0}^\infty$ is $\large \frac{1+z}{(1-z)^3}$, then $a_3-a_0$ is equal to ___________ .Combinatoryhttp://gateoverflow.in/118392/gate2017-2-47Tue, 14 Feb 2017 07:15:02 +0000#first course in probability #sheldon_ross
http://gateoverflow.in/117798/%23first-course-in-probability-%23sheldon_ross
In how many ways can 3 boys and 3 girls can sit together such that ONLY boys should together?Combinatoryhttp://gateoverflow.in/117798/%23first-course-in-probability-%23sheldon_rossFri, 10 Feb 2017 05:19:30 +0000combinatrics
http://gateoverflow.in/116415/combinatrics
Find the number of seven digit integers with sum of the digits equal to 11 and formed by using the digits 1,2 and 3 only.<br />
<br />
ans given- 161Combinatoryhttp://gateoverflow.in/116415/combinatricsMon, 06 Feb 2017 07:05:09 +0000Mathematics
http://gateoverflow.in/114815/mathematics
In how many different ways 10 identical ball can be distributed among 3 children, if each receives atleast 2 balls and no more than 4 balls?Combinatoryhttp://gateoverflow.in/114815/mathematicsThu, 02 Feb 2017 08:26:40 +0000Recurrence relation
http://gateoverflow.in/114648/recurrence-relation
**Given an integer, n, find the smallest integer m such that is divisible by n (i.e.n, is a factor of m ) and satisfies the following properties:**<br />
<br />
1. **m** must not contain zeroes in its decimal representation.<br />
2. The sum of **m's** digits must be greater than or equal to the product of **m's** digits.<br />
Given **n**, find the number of digits in **m's** decimal representation. <br />
<br />
*Note: n is not divisible by 10.*Combinatoryhttp://gateoverflow.in/114648/recurrence-relationWed, 01 Feb 2017 20:59:43 +0000Mathematics
http://gateoverflow.in/114538/mathematics
How many ways you select 20 people from 30 people and divide them into 2 group of 10 each?<br />
<br />
Create team A & B which is selected from 20 people thpse selected from 30 cricketers?<br />
<br />
<br />
<br />
is there any differnce between them?<br />
<br />
please help in thisCombinatoryhttp://gateoverflow.in/114538/mathematicsWed, 01 Feb 2017 16:43:23 +0000Mathmatics problem
http://gateoverflow.in/114318/mathmatics-problem
how many 4 digit numbers possible whose sum is equal to 12 ?Combinatoryhttp://gateoverflow.in/114318/mathmatics-problemWed, 01 Feb 2017 06:43:49 +0000permutation and combination
http://gateoverflow.in/114161/permutation-and-combination
How may binary sequence of length 10 with 4 zeroes such that 2 zeroes are never together?Combinatoryhttp://gateoverflow.in/114161/permutation-and-combinationTue, 31 Jan 2017 19:37:11 +0000Permutation and combinations
http://gateoverflow.in/114048/permutation-and-combinations
<p>I am confused with Following formulaes of combination..</p>
<p><strong>1. n!/r!</strong></p>
<p><strong>2. n+k-1 C k</strong></p>
<p><strong>3. (P+1)*2^R where (P+R are total things and P are alike).</strong></p>
<p><strong>4. question such as number of ways of choosing 10 balls out of 15 balls in which 5 are red,8 are green,2 are black.</strong></p>
<p>Anybody please provide me some reference to understand these formulaes related to combination.</p>Combinatoryhttp://gateoverflow.in/114048/permutation-and-combinationsTue, 31 Jan 2017 16:02:33 +0000ME test
http://gateoverflow.in/113436/me-test
How many number of ways can 10 balls be chosen from an urn containing 10 identical green balls, 5 identical yellow balls and 3 identical blue balls are _________?Combinatoryhttp://gateoverflow.in/113436/me-testMon, 30 Jan 2017 11:56:39 +0000ME test
http://gateoverflow.in/113274/me-test
<p>The number of ways to choose 'n' items from '2n' items of which 'n' are alike and rest are unlike is _________?</p>
<p>A) 2<sup>n</sup> B) 2<sup>n</sup>-1</p>
<p>C) (2<sup>n</sup>).<sup>2n</sup>C<sub>n </sub>D) <sup>2n</sup>C<sub>n </sub></p>Combinatoryhttp://gateoverflow.in/113274/me-testMon, 30 Jan 2017 06:05:09 +0000Permutation and Combinations
http://gateoverflow.in/113040/permutation-and-combinations
<p><img alt="" height="86" src="http://gateoverflow.in/?qa=blob&qa_blobid=3917520023243175690" width="676"></p>
<p> </p>
<p>The number of ways to choose n things from 2n things of which n are alike and rest are unlike?</p>
<p> </p>Combinatoryhttp://gateoverflow.in/113040/permutation-and-combinationsSun, 29 Jan 2017 15:03:16 +0000Recursive relation
http://gateoverflow.in/112945/recursive-relation
<p><img alt="" src="http://gateoverflow.in/?qa=blob&qa_blobid=4839287677537828555"></p>
<p>What does this problem tells and how it is related to Fibonacci series? Couldn't understand the question</p>Combinatoryhttp://gateoverflow.in/112945/recursive-relationSun, 29 Jan 2017 12:38:58 +0000test series
http://gateoverflow.in/111935/test-series
<p><img alt="" height="66" src="http://gateoverflow.in/?qa=blob&qa_blobid=360964218097595545" width="644"></p>Combinatoryhttp://gateoverflow.in/111935/test-seriesFri, 27 Jan 2017 12:31:43 +0000Discrete Mathematics
http://gateoverflow.in/111529/discrete-mathematics
consider a bit string of length 10 containing only o and 1. The number of string contain 3 consecutive 0's or 3 consecutive 1's are______Combinatoryhttp://gateoverflow.in/111529/discrete-mathematicsThu, 26 Jan 2017 09:06:28 +0000Testbook [Counting]
http://gateoverflow.in/110810/testbook-counting
<p><em><img alt="" src="http://gateoverflow.in/?qa=blob&qa_blobid=5113903718118740603"></em></p>
<p><em>I applied the formula for onto function and got 36. How are they getting 360?</em></p>Combinatoryhttp://gateoverflow.in/110810/testbook-countingWed, 25 Jan 2017 07:28:15 +0000testbook
http://gateoverflow.in/110468/testbook
no of strings upto length 3 on alphabet ∑={a,b,c,d} are (including string of zero length )??Combinatoryhttp://gateoverflow.in/110468/testbookTue, 24 Jan 2017 13:07:20 +0000summation series
http://gateoverflow.in/107156/summation-series
<p>what is the summation of this series?</p>
<p>S=<sup>n</sup>C<sub>0</sub>*2<sup>0</sup>+<sup>n</sup>C<sub><span style="font-size:10.8333px">1</span></sub>*2<sup><span style="font-size:10.8333px">1</span></sup><span style="font-size:10.8333px">+<sup>n</sup></span>C<sub><span style="font-size:10.8333px">2</span></sub>*2<sup><span style="font-size:10.8333px">2</span></sup><span style="font-size:10.8333px">+..............<sup>n</sup></span>C<sub><span style="font-size:10.8333px">n</span></sub>*2<sup><span style="font-size:10.8333px">n</span></sup></p>Combinatoryhttp://gateoverflow.in/107156/summation-seriesTue, 17 Jan 2017 17:37:40 +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 +0000rosen
http://gateoverflow.in/106117/rosen
Find a recurrence relation for the number of bit strings<br />
of length n that contain the string 01.<br />
<br />
I am getting a recurrence like An = 2^(n-2) + 2A(n-1) - A (N-2) .Answer is not given for this question.Please help and explain your steps.Combinatoryhttp://gateoverflow.in/106117/rosenMon, 16 Jan 2017 05:22:02 +0000Recurrence
http://gateoverflow.in/106020/recurrence
An organism is born on day K=1 with 1 cell. During day K=2,3,...... the organism produces $\frac{K^{2}}{K-1}$ times more new cells than it produced on day K-1. Number of cells in the organism at the end of 9th day if no cell dies is ___________<br />
<br />
My Issue: Unable to solve the recurrence relation.Combinatoryhttp://gateoverflow.in/106020/recurrenceSun, 15 Jan 2017 16:58:36 +0000