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GATE CSE
Combinatorics
Discrete Mathematics
Previous Years Questions
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Marks 1
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The number of arrangements of six identical balls in three identical bins is ___________.
GATE CSE 2022
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The number of $$4$$ digit numbers having their digits in non-decreasing order (from left to right) constructed by using ...
GATE CSE 2015 Set 3
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The recurrence relation capturing the optional execution time of the Towers of Hanoi problem with $$n$$ discs is
GATE CSE 2012
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In a class of 200 students, 125 students have taken Programming Language course, 85 students have taken Data Structures ...
GATE CSE 2004
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$$m$$ identical balls are to be placed in $$n$$ distinct bags. You are given that $$m \ge kn$$, where $$k$$ is a natural...
GATE CSE 2003
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$$n$$ couples are invited to a party with the condition that every husband should be accompanied by his wife. However, ...
GATE CSE 2003
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Let $$A$$ be a sequence of $$8$$ distinct integers sorted in ascending order. How many distinct pairs of sequence, $$B$$...
GATE CSE 2003
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The minimum number of colors required to color the vertices of a cycle with $$n$$ nodes in such a way that no two adjace...
GATE CSE 2002
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The solution to the recurrence equation $$T\left( {{2^k}} \right)$$ $$ = 3T\left( {{2^{k - 1}}} \right) + 1$$, $$T\lef...
GATE CSE 2000
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The minimum number of cards to be dealt from an arbitrarily shuffled deck of 52 cards to guarantee that three cards are ...
GATE CSE 2000
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The number of binary strings of $$n$$ zeros and $$k$$ ones such that no two ones are adjacent is:
GATE CSE 1999
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Marks 2
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There are 6 jobs with distinct difficulty levels, and 3 computers with distinct processing speeds. Each job is assigned ...
GATE CSE 2021 Set 1
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The number of permutations of the characters in LILAC so that no character appears in its original position, if the two ...
GATE CSE 2020
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The coefficient of $${x^{12}}$$ in $${\left( {{x^3} + {x^4} + {x^5} + {x^6} + ...} \right)^3}\,\,\,\,\,\,$$ is _________...
GATE CSE 2016 Set 1
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The number of distinct positive integral factors of 2014 is _______ .
GATE CSE 2014 Set 2
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A pennant is a sequence of numbers, each number being 1 or 2. An n-pennant is a sequence of numbers with sum equal to n....
GATE CSE 2014 Set 1
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There are 5 bags labeled 1 to 5. All the coins in given bag have the same weight. Some bags have coins of weight 10 gm, ...
GATE CSE 2014 Set 1
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When $$n = {2^{2k}}$$ for some $$k \ge 0$$, the recurrence relation $$$T\left( n \right) = \sqrt 2 T\left( {n/2} \right...
GATE CSE 2008
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In how many ways can $$b$$ blue balls and $$r$$ red balls be distributed in $$n$$ distinct boxes?
GATE CSE 2008
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The exponent of $$11$$ in the prime factorization of $$300!$$ is
GATE CSE 2008
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Let $${x_n}$$ denote the number of binary strings of length $$n$$ that contains no consecutive $$0s$$. Which of the foll...
GATE CSE 2008
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Let $${x_n}$$ denote the number of binary strings of length $$n$$ that contain no consecutive $$0s$$. The value of $${x_...
GATE CSE 2008
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Suppose that a robot is placed on the Cartesian plane. At each step it is allowed to move either one unit up or one unit...
GATE CSE 2007
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Suppose that a robot is placed on the Cartesian plane. At each step it is allowed to move either one unit up or one unit...
GATE CSE 2007
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What is the cardinality of the set of integers $$X$$ defined below? $$X = $$ {$$n\left| {1 \le n \le 123,\,\,\,\,\,n} \r...
GATE CSE 2006
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For each elements in a set of size $$2n$$, an unbiased coin in tossed. The $$2n$$ coin tosses are independent. An elemen...
GATE CSE 2006
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Consider the polynomial $$P\left( x \right) = {a_0} + {a_1}x + {a_2}{x^2} + {a_3}{x^3},$$ where $${a_i} \ne 0,\forall i...
GATE CSE 2006
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Let $$n = {p^2}q,$$ where $$p$$ and $$q$$ are distinct prime numbers. How many numbers $$m$$ satisfy $$1 \le m \le n$$ a...
GATE CSE 2005
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Let $$G\left( x \right) = 1/\left( {1 - x} \right)2 = \sum\limits_{i = 0}^\infty {g\left( i \right)\,{x^1}} \,\,\,,$$ ...
GATE CSE 2005
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What is the minimum number of ordered pairs of non-negative numbers that should be chosen to ensure that there are two p...
GATE CSE 2005
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The recurrence equation $$\,\,\,\,\,\,\,T\left( 1 \right) = 1$$ $$\,\,\,\,\,\,T\left( n \right) = 2T\left( {n - 1} \rig...
GATE CSE 2004
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In how many ways can we distribute 5 distinct balls, $${B_1},{B_2},......,{B_5}$$ in 5 distinct cells, $${C_1},{C_2},......
GATE CSE 2004
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Mala has a colouring book in which each English letter is drawn two times. She wants to paint each of these 52 prints wi...
GATE CSE 2004
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How many 4 digit even numbers have all 4 digits distinct?
GATE CSE 2001
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Two girls have picked 10 roses, 15 sunflowers and 14 daffodils. What is the number of ways they can divide the flowers a...
GATE CSE 1999
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Solve the following recurrence relation $$\,\,\,\,\,\,\,{x_n} = 2{x_{n - 1}} - 1\,\,n > 1$$ $$\,\,\,\,\,\,\,{x_1} = ...
GATE CSE 1998
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In a room containing 28 people, there are 18 people who speak English, 15 people who speak Hindi and 22 people who speak...
GATE CSE 1998
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The recurrence relation $$\,\,\,\,\,$$ $$T\left( 1 \right) = 2$$ $$T\left( n \right) = 3T\left( {{n \over 4}} \right)...
GATE CSE 1996
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The number of substrings (of all length inclusive) that can be formed from a character string of length $$n$$ is
GATE CSE 1994
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How many sub strings can be formed from a character string of length $$n$$?
GATE CSE 1989
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Solve the recurrence equations: $$\,\,\,\,\,\,\,\,\,\,T\left( n \right) = \left( {{n \over 2}} \right) + 1$$ $$\,\,\,\,...
GATE CSE 1988
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(a) Solve the recurrence equations $$\,\,\,\,\,\,\,\,\,T\left( n \right) = T\left( {n - 1} \right) + n$$ $$\,\,\,\,\,\,...
GATE CSE 1987
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