Recent questions and answers in Combinatory - GATE CSE Doubts

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Explain this briefly
2^n 2^n-1 (2^n-1)-1 n^2

answered 7 hours ago in Combinatory by jayeshasawa001 (90 points) | 26 views

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Rosen-Chapter 8, Ex 8.5,Question 24
Find the probability that when a fair coin is flipped five times tails comes up exactly three times, the first and last flips come up tails, or the second and fourth flips come up heads.

asked 2 days ago in Combinatory by Kushagra गुप्ता (500 points) | 17 views

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Recurrence relation - ROSEN
$1.\ a_{k}=3a_{k-1}+4^{k-1}| a_{0}=1$ $2.\ a_{k}=4a_{k-1}-4_{k-2}+k^2| a_{0}=2,a_{1}=5$

asked 3 days ago in Combinatory by Kushagra गुप्ता (500 points) | 21 views

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Self doubt on combinatorics
How many strings are there, using 10 A's, 12 B's, 11 C's, and 15 D's, such that no A is followed by a B, and no C is followed by a D?

asked Jul 29 in Combinatory by RasMalai (14 points) | 10 views

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Kenneth Rosen(7th ed). Chapter 6. Example 16.
Each user on a computer system has a password, which is six to eight characters long, where each character is an uppercase letter or a digit. Each password must contain at least one digit. How many possible passwords are there? Answer is given in book as : P^6 + P ... on for P^7 and P8. My question is why can't we calculate P^6 like 36^5 * C(6,1) * 10 ?

asked Jul 29 in Combinatory by RasMalai (14 points) | 8 views

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Sheldon Ross (8th ed). Chapter 1. Self test problems. Q 4.

asked Jul 29 in Combinatory by RasMalai (14 points) | 19 views

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Sheldon Ross(8th ed). Chapter 1. Theoretical Excercises. Q 11.

asked Jul 27 in Combinatory by RasMalai (14 points) | 21 views

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self doubt permutation and combination

answered Jun 1 in Combinatory by vps123 (9 points) | 15 views

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Self Doubt recurrence equation

asked May 31 in Combinatory by Abhipsa (27 points) | 8 views

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P&C - self doubt
Find the number of arrangements of the letters of the word "INDEPENDENCE" if they start with "P" and end with "D"

answered May 29 in Combinatory by Mohit Kumar 6 (12 points) | 21 views

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ISI Tomato Book
The value of $\sum \binom{k}{i} \binom{M-k}{100-i} [(k-i)/(M-100)]/ \binom{M}{100}$, where M – k > 100, k > 100 and $\binom{m}{n}$= m!/{(m – n)!n!} equals (summation running from i = 0 to i = 100) (a) k/M (b) M/k (c)$k/M^{2}$ (d) $M/k^{2}$

asked May 13 in Combinatory by PSDesai09 (6 points) | 13 views

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GATE2018-46 Video Solution
The number of possible min-heaps containing each value from $\{1,2,3,4,5,6,7\}$ exactly once is _______

asked Apr 18 in Combinatory by admin (3.6k points) | 12 views

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GATE2016-1-26 Video Solution
The coefficient of $x^{12}$ in $\left(x^{3}+x^{4}+x^{5}+x^{6}+\dots \right)^{3}$ is ___________.

asked Apr 18 in Combinatory by admin (3.6k points) | 2 views

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GATE2016-1-27 Video Solution
Consider the recurrence relation $a_1 =8 , a_n =6n^2 +2n+a_{n-1}$. Let $a_{99}=K\times 10^4$. The value of $K$ is __________.

asked Apr 18 in Combinatory by admin (3.6k points) | 1 view

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GATE2018-1 Video Solution
Which one of the following is a closed form expression for the generating function of the sequence $\{a_n\}$, where $a_n = 2n +3 \text{ for all } n=0, 1, 2, \dots$? $\frac{3}{(1-x)^2}$ $\frac{3x}{(1-x)^2}$ $\frac{2-x}{(1-x)^2}$ $\frac{3-x}{(1-x)^2}$

asked Apr 18 in Combinatory by admin (3.6k points) | 1 view

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GATE2016-2-29 Video Solution
The value of the expression $13^{99}\pmod{17}$ in the range $0$ to $16$, is ________.

asked Apr 18 in Combinatory by admin (3.6k points) | 3 views

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GATE2017-2-47 Video Solution
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 ___________ .

asked Apr 18 in Combinatory by admin (3.6k points) | 2 views

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GATE2019-21 Video Solution
The value of $3^{51} \text{ mod } 5$ is _____

asked Apr 18 in Combinatory by admin (3.6k points) | 3 views

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GATE2005-44 Video Solution
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 $b \equiv d \mod 5$ $4$ $6$ $16$ $24$

asked Apr 18 in Combinatory by admin (3.6k points) | 3 views

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GATE2015-3-5 Video Solution
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 ________.

asked Apr 18 in Combinatory by admin (3.6k points) | 4 views

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GATE2004-75 Video Solution
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? $9$ $8$ $7$ $6$

asked Apr 18 in Combinatory by admin (3.6k points) | 1 view

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GATE1999-2.2 Video Solution
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

asked Apr 18 in Combinatory by admin (3.6k points) | 2 views

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GATE2003-4 Video Solution
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 $A$ $2$ $30$ $56$ $256$

asked Apr 18 in Combinatory by admin (3.6k points) | 4 views

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GATE2007-84 Video Solution
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 ... $(10,10)$ starting from the initial position $(0,0)$? $^{20}\mathrm{C}_{10}$ $2^{20}$ $2^{10}$ None of the above

asked Apr 18 in Combinatory by admin (3.6k points) | 2 views

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GATE1998-1.23 Video Solution
How many sub strings of different lengths (non-zero) can be formed from a character string of length $n$? $n$ $n^2$ $2^n$ $\frac{n(n+1)}{2}$

asked Apr 18 in Combinatory by admin (3.6k points) | 1 view

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GATE2001-2.1 Video Solution
How many $4$-digit even numbers have all $4$ digits distinct $2240$ $2296$ $2620$ $4536$

asked Apr 18 in Combinatory by admin (3.6k points) | 1 view

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GATE2016-1-2 Video Solution
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 $a_n$? $a_n = a_{n-1}+ 2a_{n-2}$ $a_n = a_{n-1}+ a_{n-2}$ $a_n = 2a_{n-1}+ a_{n-2}$ $a_n = 2a_{n-1}+ 2a_{n-2}$

asked Apr 18 in Combinatory by admin (3.6k points) | 1 view

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GATE2019-5 Video Solution
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

asked Apr 18 in Combinatory by admin (3.6k points) | 2 views

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GATE2000-1.1 Video Solution
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 $3$ $8$ $9$ $12$

asked Apr 18 in Combinatory by admin (3.6k points) | 1 view

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GATE2004-IT-35 Video Solution
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? $44$ $96$ $120$ $3125$

asked Apr 18 in Combinatory by admin (3.6k points) | 2 views

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GATE2014-1-49 Video Solution
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 ${(1)}$ ... $(1,2)$ is not the same as the pennant $(2,1)$. The number of $10-$pennants is________

asked Apr 18 in Combinatory by admin (3.6k points) | 2 views

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GATE2003-34 Video Solution
$m$ 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 $k$ ... $\left( \begin{array}{c} m - kn + n + k - 2 \\ n - k \end{array} \right)$

asked Apr 18 in Combinatory by admin (3.6k points) | 5 views

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GATE2003-5 Video Solution
$n$ 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 \(^{2n}\mathrm{C}_n\times 2^n\) \(3^n\) \(\frac{(2n)!}{2^n}\) \(^{2n}\mathrm{C}_n\)

asked Apr 18 in Combinatory by admin (3.6k points) | 3 views

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GATE2008-IT-25 Video Solution
In how many ways can $b$ blue balls and $r$ red balls be distributed in $n$ distinct boxes? $\frac{(n+b-1)!\,(n+r-1)!}{(n-1)!\,b!\,(n-1)!\,r!}$ $\frac{(n+(b+r)-1)!}{(n-1)!\,(n-1)!\,(b+r)!}$ $\frac{n!}{b!\,r!}$ $\frac{(n + (b + r) - 1)!} {n!\,(b + r - 1)}$

asked Apr 18 in Combinatory by admin (3.6k points) | 1 view

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GATE1987-10b Video Solution
What is the generating function $G(z)$ for the sequence of Fibonacci numbers?

asked Apr 18 in Combinatory by admin (3.6k points) | 11 views

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GATE2015-1-26 Video Solution
$\sum\limits_{x=1}^{99}\frac{1}{x(x+1)}$ = __________________.

asked Apr 18 in Combinatory by admin (3.6k points) | 2 views

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GATE1999-1.3 Video Solution
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

asked Apr 18 in Combinatory by admin (3.6k points) | 10 views

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GATE2007-85 Video Solution
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 $(i,j + 1)$. Suppose that the robot is not allowed to traverse the line ... $^{20}\mathrm{C}_{10} - ^{8}\mathrm{C}_{4}\times ^{11}\mathrm{C}_{5}$

asked Apr 18 in Combinatory by admin (3.6k points) | 1 view

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GATE1994-1.15 Video Solution
The number of substrings (of all lengths inclusive) that can be formed from a character string of length $n$ is $n$ $n^2$ $\frac{n(n-1)}{2}$ $\frac{n(n+1)}{2}$

asked Apr 18 in Combinatory by admin (3.6k points) | 1 view

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GATE2005-50 Video Solution
Let $G(x) = \frac{1}{(1-x)^2} = \sum\limits_{i=0}^\infty g(i)x^i$, where $|x| < 1$. What is $g(i)$? $i$ $i+1$ $2i$ $2^i$

asked Apr 18 in Combinatory by admin (3.6k points) | 2 views

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