that contain at least one occurrence of two consecutive symbols, that is, "aa", "bb" or "cc". The number of such strings of length 5 that are possible is _______. (Answer in integer)

Numerical answer

Your answer

Source: GATE CSE 2025 Set 1 (CS1), IIT Roorkee. Official question paper.

GATE CSE 2023 | Question 15

Combinatorics

MCQ1 mark

The Lucas sequence

L_n

is defined by the recurrence relation:

L_n = L_{n-1} + L_{n-2}, \quad \text{for } n \ge 3,

with

L_1 = 1

and

L_2 = 3

. Which one of the options given is TRUE?

Single correct

Answer key coming soon

Source: GATE CSE 2023, IIT Kanpur. Official question paper.

GATE CSE 2023 | Question 48

Combinatorics

MCQ2 marks

Let

U = \{1, 2, \ldots, 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

with

|A| = |B| = k

and

A \cap B = \emptyset

. We say that a permutation of

U

separates

A

from

B

if one of the following is true. - All members of

A

appear in the permutation before any of the members of

B

. - All members of

B

appear in the permutation before any of the members of

A

. How many permutations of

U

separate

A

from

B

Single correct

Answer key coming soon

Source: GATE CSE 2023, IIT Kanpur. Official question paper.

GATE CSE 2022 | Question 32

Combinatorics

NAT1 mark

The number of arrangements of six identical balls in three identical bins is __________

Numerical answer

Your answer

Answer key coming soon

Source: GATE CSE 2022, IIT Kharagpur. Official question paper.

GATE CSE 2022 | Question 36

Combinatorics

MCQ2 marks

Which one of the following is the closed form for the generating function of the sequence

\{a_n\}_{n \ge 0}

defined below?

a_n = \begin{cases} n + 1, & n \text{ is odd} \\ 1, & \text{otherwise} \end{cases}

Single correct

Answer key coming soon

Source: GATE CSE 2022, IIT Kharagpur. Official question paper.

GATE CSE 2021 | Set 1 | Question 19

Combinatorics

NAT1 mark

There are 6 jobs with distinct difficulty levels, and 3 computers with distinct processing speeds. Each job is assigned to a computer such that: - The fastest computer gets the toughest job and the slowest computer gets the easiest job. - Every computer gets at least one job. The number of ways in which this can be done is __________.

Numerical answer

Your answer

Answer key coming soon

Source: GATE CSE 2021 Set 1 (CS1), IIT Bombay. Official question paper.

GATE CSE 2021 | Set 2 | Question 50

Combinatorics·Sets & Relations

NAT2 marks

Let

S

be a set consisting of 10 elements. The number of tuples of the form

(A, B)

such that

A

and

B

are subsets of

S

, and

A \subseteq B

is ________.

Numerical answer

Your answer

Answer key coming soon

Source: GATE CSE 2021 Set 2 (CS2), IIT Bombay. Official question paper.

GATE CSE 2020 | Question 42

Combinatorics

NAT2 marks

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 _________.

Numerical answer

Your answer

Answer key coming soon

Source: GATE CSE 2020, IIT Delhi. Official question paper.

GATE CSE 2019 | Question 5

Combinatorics·Sets & Relations

MCQ1 mark

Let

U = \{1, 2, \ldots, n\}

. Let

A = \{(x, X) \mid x \in X, X \subseteq U\}

. Consider the following two statements on

|A|

. I.

|A| = n2^{n-1}

II.

|A| = \sum_{k=1}^{n} k\binom{n}{k}

Which of the above statements is/are TRUE?

Single correct

Source: GATE CSE 2019, IIT Madras. Official question paper and answer key.

GATE CSE 2019 | Question 21

Combinatorics

NAT1 mark

The value of

3^{51} \bmod 5

is _____.

Numerical answer

Your answer

Source: GATE CSE 2019, IIT Madras. Official question paper and answer key.

GATE CSE 2018 | Question 1

Combinatorics

MCQ1 mark

Which one of the following is a closed form expression for the generating function of the sequence

\{a_n\}

, where

a_n = 2n + 3

for all

n = 0, 1, 2, \ldots

Single correct

Source: GATE CSE 2018, IIT Guwahati. Official question paper and answer key.

GATE CSE 2018 | Question 46

Heaps·Combinatorics

NAT2 marks

The number of possible min-heaps containing each value from {1, 2, 3, 4, 5, 6, 7} exactly once is _____.

Numerical answer

Your answer

Source: GATE CSE 2018, IIT Guwahati. Official question paper and answer key.

GATE CSE 2017 | Set 1 | Question 47

Combinatorics

NAT2 marks

The number of integers between 1 and 500 (both inclusive) that are divisible by 3 or 5 or 7 is ________.

Numerical answer

Your answer

Answer key coming soon

Source: GATE CSE 2017 Set 1, IIT Roorkee. Official question paper.

GATE CSE 2017 | Set 2 | Question 47

Combinatorics

NAT2 marks

If the ordinary generating function of a sequence

\{a_n\}_{n=0}^{\infty}

\dfrac{1 + z}{(1 - z)^3}

, then

a_3 - a_0

is equal to ________.

Numerical answer

Your answer

Answer key coming soon

Source: GATE CSE 2017 Set 2, IIT Roorkee. Official question paper.

GATE CSE 2016 | Set 1 | Question 2

Combinatorics

MCQ1 mark

Let

a_n

be the number of

n

-bit strings that do NOT contain two consecutive 1s. Which one of the following is the recurrence relation for

a_n

Single correct

Answer key coming soon

Source: GATE CSE 2016 Set 1, IISc Bengaluru. Official question paper.

GATE CSE 2016 | Set 1 | Question 26

Combinatorics

NAT2 marks

The coefficient of

x^{12}

(x^3 + x^4 + x^5 + x^6 + \cdots)^3

is ________.

Numerical answer

Your answer

Answer key coming soon

Source: GATE CSE 2016 Set 1, IISc Bengaluru. Official question paper.

GATE CSE 2016 | Set 1 | Question 27

Combinatorics·Recurrences

NAT2 marks

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 ________.

Numerical answer

Your answer

Answer key coming soon

Source: GATE CSE 2016 Set 1, IISc Bengaluru. Official question paper.

GATE CSE 2016 | Set 2 | Question 40

Trees & BST·Combinatorics

NAT2 marks

The number of ways in which the numbers 1, 2, 3, 4, 5, 6, 7 can be inserted in an empty binary search tree, such that the resulting tree has height 6, is ________. *Note: The height of a tree with a single node is 0.*

Numerical answer

Your answer

Answer key coming soon

Source: GATE CSE 2016 Set 2, IISc Bengaluru. Official question paper.

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