# Recent questions and answers in Combinatory

1
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 $3^N$ $3^N - 1$ $2^N - 1$ $2$
2
$n$ 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 $^{2n}\mathrm{C}_n\times 2^n$ $3^n$ $\frac{(2n)!}{2^n}$ $^{2n}\mathrm{C}_n$
3
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 $A$ $2$ $30$ $56$ $256$
4
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
5
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
6
How many sub strings of different lengths (non-zero) can be formed from a character string of length $n$? $n$ $n^2$ $2^n$ $\frac{n(n+1)}{2}$
7
The number of substrings (of all lengths inclusive) that can be formed from a character string of length $n$ is $n$ $n^2$ $\frac{n(n-1)}{2}$ $\frac{n(n+1)}{2}$
8
How many distinct ways are there to split $50$ identical coins among three people so that each person gets at least $5$ coins? $3^{35}$ $3^{50}-2^{50}$ $\binom{35}{2}$ $\binom{50}{15} \cdot 3^{35}$ $\binom{37}{2}$
9
Provide short answers to the following questions: 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.
10
Hello can anyone suggest good video/book to learn generating functions from?..i tried the nptel lecture..it has some audio lag. and i could not make much out of it..I am well versed in combinatorics but my calculus is weak.. Please suggest some resource that teaches generating functions from scratch
11
Which one of the following is a closed form expression for the generating function of the sequence $\{a_n\}$, where $a_n = 2n +3 \text{ for all } n=0, 1, 2, \dots$? $\frac{3}{(1-x)^2}$ $\frac{3x}{(1-x)^2}$ $\frac{2-x}{(1-x)^2}$ $\frac{3-x}{(1-x)^2}$
12
A row of $10$ houses has to be painted using the colours red, blue, and green so that each house is a single colour, and any house that is immediately to the right of a red or a blue house must be green. How many ways are there to paint the houses? $199$ $683$ $1365$ $3^{10}-2^{10}$ $3^{10}$
13
Consider all possible permutations of eight distinct elements $a, b, c, d, e, f, g, h$. In how many of them, will $d$ appear before $b$? Note that $d$ and $b$ may not necessarily be consecutive.
14
Consider $30$ multiple-choice questions, each with four options of which exactly one is correct. Then the number of ways one can get only the alternate questions correctly answered is $3^{15}$ $2^{31}$ $2 \times \begin{pmatrix} 30 \\ 15 \end{pmatrix}$ $2 \times 3^{15}$
15
Let $\pi=[x_{1},x_{2},\cdots,x_{n}]$ be a permutation of $\{1,2,\cdots,n\}.$ For $k<n,$ we say that $\pi$ has its first ascent at $k$ if $x_{1}>x_{2}\cdots>x_{k}$ and $x_{k}<x_{k+1}.$ How many permutations have their first ascent at $k?$ $\binom{n}{k}-\binom{n}{(k+1)}$ $\frac{n!}{k!}-\frac{n!}{(k+1)!}$ $\frac{n!}{(k+1)!}-\frac{n!}{(k+2)!}$ $\binom{n}{(k+1)}-\binom{n}{(k+2)}$
16
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 ______.
17
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? $9$ $8$ $7$ $6$
18
How many $4$-digit even numbers have all $4$ digits distinct $2240$ $2296$ $2620$ $4536$
19
The coefficient of $x^{12}$ in $\left(x^{3}+x^{4}+x^{5}+x^{6}+\dots \right)^{3}$ is ___________.
20
Find the solution of the recurrence relation $a_{n} = 4a_{n-1} - 3a_{n-2} + 2^{n} + n + 3\:\text{with}\: a_{0} = 1\:\text{and}\: a_{1} = 4.$
21
Find all solutions of the recurrence relation $a_{n} = 4a_{n-1} - 4a_{n-2} + (n + 1)2^{n}.$
22
In how many ways can a given positive integer $n \geq 2$ be expressed as the sum of $2$ positive integers (which are not necessarily distinct). For example, for $n=3$, the number of ways is $2$, i.e., $1+2, 2+1$. Give only the answer ... positive integer $n \geq k$ be expressed as the sum of $k$ positive integers (which are not necessarily distinct). Give only the answer without explanation.
23
Give a big-O estimate for the function $f$ in question $10$ if $f$ is an increasing function.
24
Find the characteristic roots of the linear homogeneous recurrence relation $a_{n} = 2a_{n-1} - 2a_{n-2}.$ [Note: These are complex numbers.] Find the solution of the recurrence relation in part $(A)$ with $a_{0} = 1\:\text{and}\: a_{1} = 2.$
25
Give a big-O estimate for the function $f$ given below if $f$ is an increasing function. $f (n) = 2f (n/3) + 4 \:\text{with}\: f (1) = 1.$
26
What is the minimum number of ordered pairs of non-negative numbers that should be chosen to ensure that there are two pairs $(a,b)$ and $(c,d)$ in the chosen set such that, $a \equiv c\mod 3$ and $b \equiv d \mod 5$ $4$ $6$ $16$ $24$
27
The minimum number of cards to be dealt from an arbitrarily shuffled deck of $52$ cards to guarantee that three cards are from same suit is $3$ $8$ $9$ $12$
28
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? $44$ $96$ $120$ $3125$
29
$m$ identical balls are to be placed in $n$ distinct bags. You are given that $m \geq kn$, where $k$ is a natural number $\geq 1$. In how many ways can the balls be placed in the bags if each bag must contain at least $k$ ... $\left( \begin{array}{c} m - kn + n + k - 2 \\ n - k \end{array} \right)$
30
A $1 \times 1$ chessboard has one square, a $2 \times 2$ chessboard has five squares. Continuing along this fashion, what is the number of squares on the regular $8 \times 8$ chessboard? $64$ $65$ $204$ $144$ $256$
31
Find the recurrence relation satisfied by $R_{n},$ where $R_{n}$ is the number of regions that a plane is divided into by $n$ lines, if no two of the lines are parallel and no three of the lines go through the same point. Find $R_{n}$ using iteration.
1 vote
32
Find a recurrence relation for the number of bit strings of length $n$ that contain the string $01.$ 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.
33
Solve the simultaneous recurrence relations $a_{n} = 3a_{n-1} + 2b_{n-1}$ $b_{n} = a_{n-1} + 2b_{n-1}$ with $a_{0} = 1 \: \text{and}\: b_{0} = 2.$
34
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?
35
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 $P = Q - k$ $P = Q + k$ $P = Q$ $P = Q + 2k$
36
In how many ways can $b$ blue balls and $r$ red balls be distributed in $n$ distinct boxes? $\frac{(n+b-1)!\,(n+r-1)!}{(n-1)!\,b!\,(n-1)!\,r!}$ $\frac{(n+(b+r)-1)!}{(n-1)!\,(n-1)!\,(b+r)!}$ $\frac{n!}{b!\,r!}$ $\frac{(n + (b + r) - 1)!} {n!\,(b + r - 1)}$
37
Suppose that a robot is placed on the Cartesian plane. At each step it is allowed to move either one unit up or one unit right, i.e., if it is at $(i,j)$ then it can move to either $(i + 1, j)$ or $(i,j + 1)$. How many distinct paths are there for the robot to reach the point $(10,10)$ starting from the initial position $(0,0)$? $^{20}\mathrm{C}_{10}$ $2^{20}$ $2^{10}$ None of the above
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.
Solve the recurrence relation for the number of rounds in the tournament described in question $14.$
How many rounds are in the elimination tournament described in question $14$ when there are $32$ teams?