Combinatrics and Miscellaneous topics Quiz

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 ________ .


Note - This question was Numerical Type.

Consider the recurrence relation a₁ = 8, a_n = 6n² + 2n + a_n-1. Let a₉₉ = k x 10⁴. The value of K is _____

 
Note : This question was asked as Numerical Answer Type.

Let ∑ = {a, b, c, d, e} be an alphabet. We define an encoding scheme as follows :

g(a) = 3, g(b) = 5, g(c) = 7, g(d) = 9, g(e) = 11.

Let p_i denote the i-th prime number (p₁ = 2).

For a non empty string s = a₁, a₂, ..., a_n, where each a_i ∈ ∑, define

For a non-empty sequence ⟨s_j, …, s_n⟩ of strings from Σ⁺, define 

Which of the following numbers is the encoding, h, of a non-empty sequence of strings?

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 ___________.

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 ____.

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 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 ?
 

The number of distinct positive integral factors of 2014 is _________________________

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

The number of arrangements of six identical balls in three identical bins is______. 

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  with |A| = |B| = k and A ∩ B = ∅.  We say that a permutation of U  separates A from B  if one of the following is true.
1. All members of A appear in the permutation before any of the members of B.
2. All members of B appear in the permutation before any of the members of A. 

How many permutations of U separate A from B?