Combinatorics | GATE PYQ [2010-2025]

Last Updated :

Discuss

Comments

Question 1

Each of the nine words in the sentence

"The quick brown fox jumps over the lazy dog" is written on a separate piece of paper. These nine pieces of paper are kept in a box. One of the pieces is drawn at random from the box. The expected length of the word drawn is _____________.

(The answer should be rounded to one decimal place.)  [ NAT || GATE 2014 Set-2 || 1 Marks ]

• 3.9

Question 2

The number of distinct positive integral factors of 2014 is _________ [ NAT || GATE 2014 Set-2 || 2 Marks ]

• 8

Question 3

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)}, the set of all possible 2-pennants is {(2), (1,1)}and the set of all 3-pennants is {(2,1), (1,1,1), (1,2)}.

Note that the pennant (1,2) is not the same as the pennant (2,1). The number of 10-pennants is ______________. [ NAT || GATE 2014 Set-1 || 2 Marks ]

• 89

Question 4

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 _____. [ NAT || GATE 2015 Set-3 || 1 Marks ]

• 15

Question 5

The number of divisors of 2100 is ______. [ NAT || GATE 2015 Set-2 || 1 Marks ]

• 36

Question 6

Σ₁⁹⁹ 1/x(x+1) = ______. [ NAT || GATE 2015 Set-1 || 2 Marks ]

• 0.99

Question 7

Let a_n represent the number of bit strings of length n containing two consecutive 1s. What is the recurrence relation for a_n ? [ MCQ || GATE 2015 Set-1 || 2 Marks ]

• a_n–2 + a_n–1 + 2^n–2

• a_n–2 + 2a_n–1 + 2^n–2

• a_n–2 + a_n–1 + 2^n–2

• a_n–2 + 2a_n–1 + 2^n–2

Question 8

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 ? [ MCQ || GATE 2016 Set-1 || 1 Marks ]

• 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

Question 9

The coefficient of x ¹² in (x ³ + x ⁴ + x ⁵ + x ⁶ +...)³ is _________. [ NAT || GATE 2016 Set-1 || 2 Marks ]

• 10

Question 10

Consider the recurrence relation a₁ = 8, a_n = 6n ² + 2n + a _n-1 . Let a ⁹⁹ = K × 10 ⁴ . The value of K is ___________. [ NAT || GATE 2016 Set-1 || 2 Marks ]

• 198

Tags:

GATE CS

GATE

Discrete Mathematics