Combinatorics · Discrete Mathematics · GATE CSE

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Marks 1

GATE CSE 2022
The number of arrangements of six identical balls in three identical bins is ___________.
GATE CSE 2021 Set 1
There are 6 jobs with distinct difficulty levels, and 3 computers with distinct processing speeds. Each job is assigned to a computer such that: - Th...
GATE CSE 2015 Set 3
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 se...
GATE CSE 2012
The recurrence relation capturing the optional execution time of the Towers of Hanoi problem with $$n$$ discs is
GATE CSE 2004
In a class of 200 students, 125 students have taken Programming Language course, 85 students have taken Data Structures course, 65 student have taken ...
GATE CSE 2003
$$m$$ identical balls are to be placed in $$n$$ distinct bags. You are given that $$m \ge kn$$, where $$k$$ is a natural number $$ \ge 1$$. In how man...
GATE CSE 2003
$$n$$ 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...
GATE CSE 2003
Let $$A$$ be a sequence of $$8$$ distinct integers sorted in ascending order. How many distinct pairs of sequence, $$B$$ and $$C$$ are there such that...
GATE CSE 2002
The minimum number of colors required to color the vertices of a cycle with $$n$$ nodes in such a way that no two adjacent nodes have the same colour...
GATE CSE 2000
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
GATE CSE 2000
The solution to the recurrence equation $$T\left( {{2^k}} \right)$$ $$ = 3T\left( {{2^{k - 1}}} \right) + 1$$, $$T\left( 1 \right) = 1$$ is:...
GATE CSE 1999
The number of binary strings of $$n$$ zeros and $$k$$ ones such that no two ones are adjacent is:

Marks 2

GATE CSE 2023
Let $$U = \{ 1,2,....,n\} $$, where n is a large positive integer greater than 1000. Let k be a positive integer less than n. Let A, B be subsets of U...
GATE CSE 2020
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 ...
GATE CSE 2016 Set 1
The coefficient of $${x^{12}}$$ in $${\left( {{x^3} + {x^4} + {x^5} + {x^6} + ...} \right)^3}\,\,\,\,\,\,$$ is _____________.
GATE CSE 2014 Set 2
The number of distinct positive integral factors of 2014 is _______ .
GATE CSE 2014 Set 1
There are 5 bags labeled 1 to 5. All the coins in given bag have the same weight. Some bags have coins of weight 10 gm, other have coins of weight 11 ...
GATE CSE 2014 Set 1
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-p...
GATE CSE 2008
Let $${x_n}$$ denote the number of binary strings of length $$n$$ that contains no consecutive $$0s$$. Which of the following recurrences does $${x_n...
GATE CSE 2008
When $$n = {2^{2k}}$$ for some $$k \ge 0$$, the recurrence relation $$$T\left( n \right) = \sqrt 2 T\left( {n/2} \right) + \sqrt n ,\,\,T\left( 1 \ri...
GATE CSE 2008
In how many ways can $$b$$ blue balls and $$r$$ red balls be distributed in $$n$$ distinct boxes?
GATE CSE 2008
The exponent of $$11$$ in the prime factorization of $$300!$$ is
GATE CSE 2008
Let $${x_n}$$ denote the number of binary strings of length $$n$$ that contain no consecutive $$0s$$. The value of $${x_5}$$ is
GATE CSE 2007
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 $$(i...
GATE CSE 2007
Suppose that a robot is placed on the Cartesian plane. At each step it is allowed to move either one unit up or one unit right, i.e., if it is at $$(i...
GATE CSE 2006
Consider the polynomial $$P\left( x \right) = {a_0} + {a_1}x + {a_2}{x^2} + {a_3}{x^3},$$ where $${a_i} \ne 0,\forall i$$. The minimum number of mult...
GATE CSE 2006
For each elements in a set of size $$2n$$, an unbiased coin in tossed. The $$2n$$ coin tosses are independent. An element is chhoosen if the correspon...
GATE CSE 2006
What is the cardinality of the set of integers $$X$$ defined below? $$X = $$ {$$n\left| {1 \le n \le 123,\,\,\,\,\,n} \right.$$ is not divisible by ...
GATE CSE 2005
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 $$(c, d)...
GATE CSE 2005
Let $$G\left( x \right) = 1/\left( {1 - x} \right)2 = \sum\limits_{i = 0}^\infty {g\left( i \right)\,{x^1}} \,\,\,,$$ where $$\left| x \right| < ...
GATE CSE 2005
Let $$n = {p^2}q,$$ where $$p$$ and $$q$$ are distinct prime numbers. How many numbers $$m$$ satisfy $$1 \le m \le n$$ and $$gcd\left( {m.n} \right) =...
GATE CSE 2004
In how many ways can we distribute 5 distinct balls, $${B_1},{B_2},......,{B_5}$$ in 5 distinct cells, $${C_1},{C_2},.....,{C_5}$$ such that Ball $${B...
GATE CSE 2004
The recurrence equation $$\,\,\,\,\,\,\,T\left( 1 \right) = 1$$ $$\,\,\,\,\,\,T\left( n \right) = 2T\left( {n - 1} \right) + n,\,n \ge 2$$ evaluate...
GATE CSE 2004
Mala has a 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 ...
GATE CSE 2001
How many 4 digit even numbers have all 4 digits distinct?
GATE CSE 1999
Two girls have picked 10 roses, 15 sunflowers and 14 daffodils. What is the number of ways they can divide the flowers among themselves?
GATE CSE 1998
Solve the following recurrence relation $$\,\,\,\,\,\,\,{x_n} = 2{x_{n - 1}} - 1\,\,n > 1$$ $$\,\,\,\,\,\,\,{x_1} = 2$$
GATE CSE 1998
In a room containing 28 people, there are 18 people who speak English, 15 people who speak Hindi and 22 people who speak Kannada, 9 persons speak both...
GATE CSE 1996
The recurrence relation $$\,\,\,\,\,$$ $$T\left( 1 \right) = 2$$ $$T\left( n \right) = 3T\left( {{n \over 4}} \right) + n$$ has the solution $$T(n...
GATE CSE 1994
The number of substrings (of all length inclusive) that can be formed from a character string of length $$n$$ is
GATE CSE 1989
How many sub strings can be formed from a character string of length $$n$$?
GATE CSE 1988
Solve the recurrence equations: $$\,\,\,\,\,\,\,\,\,\,T\left( n \right) = \left( {{n \over 2}} \right) + 1$$ $$\,\,\,\,\,\,\,\,\,\,\,T\left( 1 \right...
GATE CSE 1987
(a) Solve the recurrence equations $$\,\,\,\,\,\,\,\,\,T\left( n \right) = T\left( {n - 1} \right) + n$$ $$\,\,\,\,\,\,\,\,\,T\left( 1 \right) = 1T$$...
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