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 _______.

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, …?

The  number  of  binary  strings  of  n  zeroes  and  k  ones  that  no  two  ones  are adjacent is

Two girls have picked 10 roses, 15 sunflowers and 15 daffodils. What is the number of ways they can divide the flowers amongst themselves?

The minimum number of cards to be dealt from an arbitrarily shuffled deck of 52 cards to guarantee that three cards are from some same suit is

How many 4-digit even numbers have all 4 digits distinct?

(a) In how many ways can a given positive integer n ≥ 2 be expressed as the sum of 2 positive integers (which are not necessarily distinct). For example, for n = 3, the number of ways is 2, i.e., 1+2, 2+1. Give only the answer without any explanation.

(b) In how many ways can a given positive integer n ≥ 3 be expressed as the sum of 3 positive integers (which are not necessarily distinct). For example, for n = 4, the number of ways is 3, i.e., 1+2+1, 2+1+1. Give only the answer without any explanation.

(c) In how many ways can a given positive integer n ≥ k be expressed as the sum of k positive integers (which are not necessarily distinct)? Give only the answer without explanation.

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 by her husband. The number of different gatherings possible at the party is

m identical balls are to be placed in n distinct bags. You are given that m ≥ kn, where, k is a natural number ≥1. In how many ways can the balls be placed in the bags if each bag must contain at least k balls?

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?

Consider the sequence <x_n>, n>= 0 defined by the recurrence relation x_{n + 1} = c . x_n²- 2, where c > 0.
Suppose there exists a non-empty, open interval (a, b) such that for all x₀ satisfying a < x₀ < b, the sequence converges to a limit. The sequence converges to the value?

Consider the sequence <x_n>, n>= 0 defined by the recurrence relation x_{n + 1} = c. x_n²- 2, where c > 0.
For which of the following values of c, does there exist a non-empty open interval (a,b) such that the sequence x_n, converges for all x₀ satisfying a < x₀

(i) 0.25
(ii) 0.35
(iii) 0.45
(iv) 0.5

The exponent of 11 in the prime factorization of 300! is

In how many ways can b blue balls and r red balls be distributed in n distinct boxes?

Choose the correct alternatives (More than one may be correct).

Let , where |x|<1. What is g(i)?

If the ordinary generating function of a sequence is , then a₃ - a₀ is equal to ________.

The number of divisors of 2100 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 coefficient of x¹² in (x³ + x⁴ + x⁵ + x⁶ + ...)³ is _________.

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

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

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,j) then it can move to either (i+1,j) or (i,j+1).

How many distinct paths are there for the robot to reach the point (10,10) starting from the initial position (0,0)?

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,j) then it can move to either (i+1,j) or (i,j+1).

Suppose that the robot is not allowed to traverse the line segment from (4,4) to (5,4). With this constraint, how many distinct paths are there for the robot to reach (10,10) starting from (0,0)?

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 ________.

The number of distinct positive integral factors of 2014 is _________.