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Recent questions and answers in Combinatory

0 votes
1 answer
1
Find the least number of cables required to connect eight computers to four printers to guarantee that for every choice of four of the eight computers, these four computers can directly access four different printers. Justify your answer.
answered 6 days ago in Combinatory aditi19 92 views
8 votes
8 answers
2
The number of permutations of the characters in LILAC so that no character appears in its original position, if the two L’s are indistinguishable, is ______.
answered Jan 19 in Combinatory ayushSa 5.1k views
14 votes
5 answers
3
Let $P =\sum_{\substack{1\le i \le 2k \\ i\;odd}} i$ and $Q = \sum_{\substack{1 \le i \le 2k \\ i\;even}} i$, where $k$ is a positive integer. Then $P = Q - k$ $P = Q + k$ $P = Q$ $P = Q + 2k$
answered Jan 16 in Combinatory Surya_Dev Chaturvedi 2k views
28 votes
8 answers
4
How many $4$-digit even numbers have all $4$ digits distinct $2240$ $2296$ $2620$ $4536$
answered Jan 14 in Combinatory eshita1997 6.1k views
0 votes
1 answer
5
Suppose that a basketball league has $32$ teams, split into two conferences of $16$ teams each. Each conference is split into three divisions. Suppose that the North Central Division has five teams. Each of the teams in the North Central Division plays four games ... the other conference. In how many different orders can the games of one of the teams in the North Central Division be scheduled?
answered Jan 10 in Combinatory reboot 74 views
0 votes
1 answer
6
Suppose that a weapons inspector must inspect each of five different sites twice, visiting one site per day. The inspector is free to select the order in which to visit these sites, but cannot visit site $\text{X},$ the most suspicious site, on two consecutive days. In how many different orders can the inspector visit these sites?
answered Jan 10 in Combinatory reboot 51 views
2 votes
2 answers
7
How many ways are there to pack six copies of the same book into four identical boxes, where a box can contain as many as six books? $4$ $6$ $7$ $9$
answered Jan 10 in Combinatory reboot 154 views
0 votes
1 answer
8
Given a standard deck of cards, there $52!$ are different permutations of the cards. Given two identical standard decks of cards, how many different permutations are there?
answered Jan 10 in Combinatory reboot 124 views
36 votes
4 answers
9
Let $a_n$ be the number of $n$-bit strings that do NOT contain two consecutive $1's$. Which one of the following is the recurrence relation for $a_n$? $a_n = a_{n-1}+ 2a_{n-2}$ $a_n = a_{n-1}+ a_{n-2}$ $a_n = 2a_{n-1}+ a_{n-2}$ $a_n = 2a_{n-1}+ 2a_{n-2}$
answered Jan 3 in Combinatory reboot 5k views
1 vote
3 answers
10
Let $A$ be a set of $n$ elements. The number of ways, we can choose an ordered pair $(B,C)$, where $B,C$ are disjoint subsets of $A$, equals $n^2$ $n^3$ $2^n$ $3^n$
answered Dec 10, 2020 in Combinatory Priyansh Singh 274 views
13 votes
6 answers
11
What is the minimum number of students needed in a class to guarantee that there are at least $6$ students whose birthdays fall in the same month ? $6$ $23$ $61$ $72$ $91$
answered Nov 27, 2020 in Combinatory StoneHeart 1.2k views
17 votes
4 answers
12
Let $U = \{1, 2, \dots , n\}$ Let $A=\{(x, X) \mid x \in X, X \subseteq U \}$. Consider the following two statements on $\mid A \mid$. $\mid A \mid = n2^{n-1}$ $\mid A \mid = \Sigma_{k=1}^{n} k \begin{pmatrix} n \\ k \end{pmatrix}$ Which of the above statements is/are TRUE? Only I Only II Both I and II Neither I nor II
answered Nov 27, 2020 in Combinatory StoneHeart 5k views
50 votes
12 answers
13
The number of $4$ digit numbers having their digits in non-decreasing order (from left to right) constructed by using the digits belonging to the set $\{1, 2, 3\}$ is ________.
answered Nov 27, 2020 in Combinatory StoneHeart 7.1k views
47 votes
12 answers
14
A pennant is a sequence of numbers, each number being $1$ or $2$. An $n-$pennant is a sequence of numbers with sum equal to $n$. For example, $(1,1,2)$ is a $4-$pennant. The set of all possible $1-$pennants is ${(1)}$, the set of all possible $2-$ ... $(1,2)$ is not the same as the pennant $(2,1)$. The number of $10-$pennants is________
answered Nov 27, 2020 in Combinatory StoneHeart 4.9k views
10 votes
12 answers
15
What is the remainder when $4444^{4444}$ is divided by $9?$ $1$ $2$ $5$ $7$ $8$
answered Nov 27, 2020 in Combinatory StoneHeart 1.7k views
25 votes
5 answers
16
The coefficient of $x^{3}$ in the expansion of $(1 + x)^{3} (2 + x^{2})^{10}$ is. $2^{14}$ $31$ $\left ( \frac{3}{3} \right ) + \left ( \frac{10}{1} \right )$ $\left ( \frac{3}{3} \right ) + 2\left ( \frac{10}{1} \right )$ $\left ( \frac{3}{3} \right ) \left ( \frac{10}{1} \right ) 2^{9}$
answered Nov 27, 2020 in Combinatory StoneHeart 1.7k views
44 votes
9 answers
17
If the ordinary generating function of a sequence $\left \{a_n\right \}_{n=0}^\infty$ is $\large \frac{1+z}{(1-z)^3}$, then $a_3-a_0$ is equal to ___________ .
answered Nov 27, 2020 in Combinatory StoneHeart 9.3k views
45 votes
15 answers
18
The coefficient of $x^{12}$ in $\left(x^{3}+x^{4}+x^{5}+x^{6}+\dots \right)^{3}$ is ___________.
answered Nov 27, 2020 in Combinatory StoneHeart 13.9k views
19 votes
8 answers
19
Let $G(x) = \frac{1}{(1-x)^2} = \sum\limits_{i=0}^\infty g(i)x^i$, where $|x| < 1$. What is $g(i)$? $i$ $i+1$ $2i$ $2^i$
answered Nov 27, 2020 in Combinatory StoneHeart 3.5k views
20 votes
5 answers
20
It is required to divide the $2n$ members of a club into $n$ disjoint teams of $2$ members each. The teams are not labelled. The number of ways in which this can be done is: $\frac{\left ( 2n \right )!}{2^{n}}$ $\frac{\left ( 2n \right )!}{n!}$ $\frac{\left ( 2n \right )!}{2^n . n!}$ $\frac{n!}{2}$ None of the above.
answered Nov 27, 2020 in Combinatory StoneHeart 2.4k views
33 votes
12 answers
21
In how many ways can we distribute $5$ distinct balls, $B_1, B_2, \ldots, B_5$ in $5$ distinct cells, $C_1, C_2, \ldots, C_5$ such that Ball $B_i$ is not in cell $C_i$, $\forall i= 1,2,\ldots 5$ and each cell contains exactly one ball? $44$ $96$ $120$ $3125$
answered Nov 27, 2020 in Combinatory StoneHeart 5.7k views
0 votes
3 answers
22
There are four bus lines between $A$ and $B$; and three bus lines between $B$ and $C$. The number of way a person roundtrip by bus from $A$ to $C$ by way of $B$ will be $12$ $7$ $144$ $264$
answered Nov 10, 2020 in Combinatory Ashwani Kumar 2 127 views
45 votes
8 answers
23
Mala has the colouring book in which each English letter is drawn two times. She wants to paint each of these $52$ prints with one of $k$ colours, such that the colour pairs used to colour any two letters are different. Both prints of a letter can also be coloured with the same colour. What is the minimum value of $k$ that satisfies this requirement? $9$ $8$ $7$ $6$
answered Nov 6, 2020 in Combinatory Dheeraj Varma 6.7k views
33 votes
2 answers
24
A line $L$ in a circuit is said to have a $stuck-at-0$ fault if the line permanently has a logic value $0$. Similarly a line $L$ in a circuit is said to have a $stuck-at-1$ fault if the line permanently has a logic value $1$. A circuit is said to have a multiple $stuck-at$ ... total number of distinct multiple $stuck-at$ faults possible in a circuit with $N$ lines is $3^N$ $3^N - 1$ $2^N - 1$ $2$
answered Oct 17, 2020 in Combinatory varunrajarathnam 3.4k views
31 votes
3 answers
25
$n$ couples are invited to a party with the condition that every husband should be accompanied by his wife. However, a wife need not be accompanied by her husband. The number of different gatherings possible at the party is \(^{2n}\mathrm{C}_n\times 2^n\) \(3^n\) \(\frac{(2n)!}{2^n}\) \(^{2n}\mathrm{C}_n\)
answered Oct 16, 2020 in Combinatory varunrajarathnam 4.5k views
34 votes
4 answers
26
Let $A$ be a sequence of $8$ distinct integers sorted in ascending order. How many distinct pairs of sequences, $B$ and $C$ are there such that each is sorted in ascending order, $B$ has $5$ and $C$ has $3$ elements, and the result of merging $B$ and $C$ gives $A$ $2$ $30$ $56$ $256$
answered Oct 16, 2020 in Combinatory varunrajarathnam 6.2k views
26 votes
5 answers
27
Two girls have picked $10$ roses, $15$ sunflowers and $15$ daffodils. What is the number of ways they can divide the flowers among themselves? $1638$ $2100$ $2640$ None of the above
answered Oct 14, 2020 in Combinatory varunrajarathnam 6.5k views
24 votes
5 answers
28
The number of binary strings of $n$ zeros and $k$ ones in which no two ones are adjacent is $^{n-1}C_k$ $^nC_k$ $^nC_{k+1}$ None of the above
answered Oct 14, 2020 in Combinatory varunrajarathnam 4.2k views
2 votes
1 answer
29
15. a) How many cards must be chosen from a standard deck of 52 cards to guarantee that at least two of the four aces are chosen? b) How many cards must be chosen from a standard deck of 52 cards to guarantee that at least two of the four aces and at least ... many cards must be chosen from a standard deck of 52 cards to guarantee that there are at least two cards of each of two different kinds?
asked Jul 4, 2020 in Combinatory Sanjay Sharma 317 views
4 votes
1 answer
30
How many pairs $(x,y)$ such that $x+y <= k$, where x y and k are integers and $x,y>=0, k > 0$. Solve by summation rules. Solve by combinatorial argument.
asked Jun 8, 2020 in Combinatory dd 421 views
0 votes
2 answers
33
Suppose that there are $n = 2^{k}$ teams in an elimination tournament, where there are $\frac{n}{2}$ games in the first round, with the $\frac{n}{2} = 2^{k-1}$ winners playing in the second round, and so on. Develop a recurrence relation for the number of rounds in the tournament.
asked May 10, 2020 in Combinatory Lakshman Patel RJIT 205 views
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