1
The number of arrangements of six identical balls in three identical bins is _____________ .
2
Which one of the following is the closed form for the generating function of the sequence $\{ a_{n} \}_{n \geq 0}$ defined below? $$a_{n} = \left\{\begin{matrix} n + 1, ... 5 answers 3 Consider the following recurrence:$$\begin{array}{} f(1) & = & 1; \\ f(2n) & = & 2f(n) – 1, & \; \text{for}\; n \geq 1; \\ f(2n+1) & = & 2f(n) + 1, & \; \text{for}\; n...
4
Let $S$ be a set of 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 ___________
5
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 ...
6
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 ______.
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Let $U = \{1, 2, \dots , n\}$ Let $A=\{(x, X) \mid x \in X, X \subseteq U \}$. Consider the following two statements on $\mid A \mid$. $\mid A \mid = n2^{n-1}$ $\mid A \m... 14 answers 8 The value of$3^{51} \text{ mod } 5$is _____ 11 answers 9 Which one of the following is a closed form expression for the generating function of the sequence$\{a_n\}$, where$a_n = 2n +3 \text{ for all } n=0, 1, 2, \dots$?$\fra...
10
If the ordinary generating function of a sequence $\left \{a_n\right \}_{n=0}^\infty$ is $\large \frac{1+z}{(1-z)^3}$, then $a_3-a_0$ is equal to ___________ .
11
How many substrings (of all lengths inclusive) can be formed from a character string of length $n$? Assume all characters to be distinct, prove your answer.
The number of ways in which $5\; A's, 5\; B's$ and $5\; C's$ can be arranged in a row is: $15!/(5!)^{3}$ $15!$ $\left(\frac{15}{5}\right)$ $15!(5!3!)$.
The number of rooted binary trees with $n$ nodes is, Equal to the number of ways of multiplying $(n+1)$ matrices. Equal to the number of ways of arranging $n$ out of $2 n... 8 answers 14 What is the generating function$G(z)$for the sequence of Fibonacci numbers? 4 answers 15 Suppose that a robot is placed on the Cartesian plane. At each step it is allowed to move either one unit up or one unit right, i.e., if it is at$(i,j)$then it can move... 10 answers 16 Consider the recurrence relation$a_1 =8 , a_n =6n^2 +2n+a_{n-1}$. Let$a_{99}=K\times 10^4$. The value of$K$is __________. 17 answers 17 The coefficient of$x^{12}$in$\left(x^{3}+x^{4}+x^{5}+x^{6}+\dots \right)^{3}$is ___________. 4 answers 18 Let$a_n$be the number of$n$-bit strings that do NOT contain two consecutive$1's$. Which one of the following is the recurrence relation for$a_n$?$a_n = a_{n-1}+ 2a_...
The value of the expression $13^{99}\pmod{17}$ in the range $0$ to $16$, 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 ___...