GATE Overflow - Recent questions in Combinatory
http://gateoverflow.in/questions/mathematics/discrete-mathematics/combinatory
Powered by Question2AnswerKenneth rosen chp 6 question 39 counting
http://gateoverflow.in/130005/kenneth-rosen-chp-6-question-39-counting
How many partial functions are there from a set with m elements to a set with n elements, where m and n are positive integers.<br />
<br />
Answer : (n+1)^m <br />
<br />
How . Can anyone please help?Combinatoryhttp://gateoverflow.in/130005/kenneth-rosen-chp-6-question-39-countingWed, 17 May 2017 14:54:04 +0000Kenneth rosen chapter 6 question 41 counting
http://gateoverflow.in/129991/kenneth-rosen-chapter-6-question-41-counting
A palindrome is a string whose reversal is identical to the string. how many bit strings of length n are palindromes ?Combinatoryhttp://gateoverflow.in/129991/kenneth-rosen-chapter-6-question-41-countingWed, 17 May 2017 13:00:29 +0000Kenneth rosen chapter 6 question 17 counting
http://gateoverflow.in/129976/kenneth-rosen-chapter-6-question-17-counting
How many strings of 5 ASCII characters contain the character @ atleast once ? [ NOTE : there are 128 ascii characters ] <br />
<br />
<br />
<br />
Answer is : 1,321,368,961<br />
<br />
Can anyone explain how ?Combinatoryhttp://gateoverflow.in/129976/kenneth-rosen-chapter-6-question-17-countingWed, 17 May 2017 11:15:26 +0000basic aptitude
http://gateoverflow.in/129883/basic-aptitude
<table>
<tbody>
<tr>
<td>
<p>12 chairs are arranged in a row and are numbered 1 to 12 4 men have to be seated in these chairs so that the chairs numbered 1 and 8 should be occupied and no two men occupy adjacent chairs.</p>
<p>Find the number of ways the task can be done.</p>
</td>
</tr>
</tbody>
</table>
<table>
<tbody>
<tr>
<td> </td>
<td><strong>A.</strong></td>
<td>
<p>360</p>
</td>
</tr>
<tr>
<td> </td>
<td><strong>B.</strong></td>
<td>
<p>384</p>
</td>
</tr>
<tr>
<td> </td>
<td><strong>C.</strong></td>
<td>
<p>432</p>
</td>
</tr>
<tr>
<td> </td>
<td><strong>D.</strong></td>
<td>
<p>470</p>
</td>
</tr>
</tbody>
</table>Combinatoryhttp://gateoverflow.in/129883/basic-aptitudeMon, 15 May 2017 14:03:54 +0000basic aptitude
http://gateoverflow.in/129878/basic-aptitude
<table>
<tbody>
<tr>
<td>
<p>The letter of the word LABOUR are permuted in all possible ways and the words thus formed are arranged as in a dictionary. What is the rank of the word LABOUR?</p>
</td>
</tr>
</tbody>
</table>
<table>
<tbody>
<tr>
<td> </td>
<td><strong>A.</strong></td>
<td>
<p>275</p>
</td>
</tr>
<tr>
<td> </td>
<td><strong>B.</strong></td>
<td>
<p>251</p>
</td>
</tr>
<tr>
<td> </td>
<td><strong>C.</strong></td>
<td>
<p>240</p>
</td>
</tr>
<tr>
<td> </td>
<td><strong>D.</strong></td>
<td>
<p>242</p>
</td>
</tr>
</tbody>
</table>
<p> can someone give me proper way to solve ?</p>Combinatoryhttp://gateoverflow.in/129878/basic-aptitudeMon, 15 May 2017 13:48:10 +0000Subsequence Theorem From Rosen.
http://gateoverflow.in/129834/subsequence-theorem-from-rosen
There is a theorem<br />
<br />
Every sequence of n^2 + 1 distinct real numbers contains a subsequence of length n + 1 that<br />
is either strictly increasing or strictly decreasing.<br />
<br />
Proof: Let a1, a2,...,an2+1 be a sequence of n^2 + 1 distinct real numbers.Associate an ordered<br />
pair with each term of the sequence, namely, associate (ik, dk) to the term ak, where ik is the<br />
length of the longest increasing subsequence starting at ak, and dk is the length of the longest<br />
decreasing subsequence starting at ak.<br />
Suppose that there are no increasing or decreasing subsequences of length n + 1. Then ik<br />
and dk are both positive integers less than or equal to n, for k = 1, 2,...,n^2 + 1. Hence, by the<br />
product rule there are n2 possible ordered pairs for (ik, dk). By the pigeonhole principle, two of<br />
these n^2 + 1 ordered pairs are equal. In other words, there exist terms as and at , with s<t<br />
such that is = it and ds = dt . We will show that this is impossible. Because the terms of the<br />
sequence are distinct, either as < at or as > at . If as < at , then, because is = it , an increasing<br />
subsequence of length it + 1 can be built starting at as, by taking as followed by an increasing<br />
subsequence of length it beginning at at . This is a contradiction. Similarly, if as > at , the same<br />
reasoning shows that ds must be greater than dt , which is a contradiction.<br />
<br />
<br />
<br />
<br />
<br />
My question is how n^2 ordered pair possible for (ik,dk) ??Combinatoryhttp://gateoverflow.in/129834/subsequence-theorem-from-rosenMon, 15 May 2017 03:12:13 +0000maths
http://gateoverflow.in/129379/maths
In how many ways three girls and nine boys can be seated int two vans each having numbered seats ,3 in the front and and 4 at the back ?<br />
<br />
How many arrangements are possible if 3 girls sit together in back row on adjacent seats?Combinatoryhttp://gateoverflow.in/129379/mathsFri, 12 May 2017 10:58:33 +0000basic aptitude
http://gateoverflow.in/129131/basic-aptitude
<table>
<tbody>
<tr>
<td>
<p>Ten different letters of alphabet are given, words with 5 letters are formed from these given letters. Then, the number of words which have at least one letter repeated is:</p>
</td>
</tr>
</tbody>
</table>
<table>
<tbody>
<tr>
<td> </td>
<td><strong>A.</strong></td>
<td>
<p>69760</p>
</td>
</tr>
<tr>
<td> </td>
<td><strong>B.</strong></td>
<td>
<p>30240</p>
</td>
</tr>
<tr>
<td> </td>
<td><strong>C.</strong></td>
<td>
<p>99748</p>
</td>
</tr>
<tr>
<td> </td>
<td><strong>D.</strong></td>
<td>
<p>42386</p>
</td>
</tr>
</tbody>
</table>
<p> </p>
<p>i got ans from the method totol words - no word repeated , but if i want to do with general method means 10*10*9*8*7 *5!/2! + 10*10*10*9*8*5!/3! + 10*10*10*10*9*5!/4! + 10*10*10*10*!0 from this getting different ans where is going wrong ? someone verify pls </p>Combinatoryhttp://gateoverflow.in/129131/basic-aptitudeWed, 10 May 2017 06:22:38 +0000#rosen , #counting , #5.1 , 41(a)
http://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 )Combinatoryhttp://gateoverflow.in/128608/%23rosen-%23counting-%235-1-41-aSun, 07 May 2017 17:51:30 +0000Discrete Mathematics Thegatebook
http://gateoverflow.in/128449/discrete-mathematics-thegatebook
how many positive integers between 50 and 100,<br />
<br />
(a) divisible by 7<br />
(b) divisible by 11<br />
(c) divisible by 7 and 11?Combinatoryhttp://gateoverflow.in/128449/discrete-mathematics-thegatebookSun, 07 May 2017 10:25:10 +0000combinatorics
http://gateoverflow.in/127196/combinatorics
In how many ways 8 different shirts can be distributed to 4 different people so that each will get 2 shirts?Combinatoryhttp://gateoverflow.in/127196/combinatoricsFri, 28 Apr 2017 09:40:08 +0000combinatorics
http://gateoverflow.in/127195/combinatorics
Number of 5 digit number having there digits in non decreasing order (from left to right) constructed by using the digits belonging to the set {1, 2, 3, 4, 5, 6, 7, 8, 9} ?Combinatoryhttp://gateoverflow.in/127195/combinatoricsFri, 28 Apr 2017 09:38:19 +0000Generalised 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 21:58:10 +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 15:46:38 +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/combinatoricsFri, 21 Apr 2017 00:26:11 +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 18:04:50 +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 22:15:20 +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 16:31:03 +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 10:23:36 +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 15:15:08 +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 14:15:26 +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-2017Thu, 06 Apr 2017 01:20:21 +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 14:57:07 +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 14:27:44 +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 14:17:03 +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 13:58:33 +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 23:38:44 +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 23:12:55 +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 23:09:06 +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 23:00:33 +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 21:19:37 +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 18:49:55 +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 13:17:51 +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 13:03:30 +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 12:56:46 +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 12:39:21 +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 11:39:27 +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 10:25:04 +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 08:58:53 +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 21:10:18 +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 16:56:36 +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 15:34:21 +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 13:47:01 +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 13:38:37 +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 11:32:11 +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 11:27:55 +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 14:54:39 +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 17:00:07 +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 20:17:51 +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 20:09:29 +0000