1
vote
3answers
69 views

Product of “reversed” numbers

Consider any 2 binary numbers, e.g.: 10101011 ; 11111101 and their product, say P. "Reverse" (mirror image) all the digits of the 2 numbers, e.g.: ...
0
votes
0answers
29 views

Count of binary matrices having number of ones in each row less than or equal to A and number of ones in each column less than or equal to B?

Example: for $N = 2, M = 2, A = 0, B = 0$, we have only one possible matrix. $$ = \left[ \begin{array}{cc} 0 & 0 \\ 0 & 0 \\ \end{array} \right] $$
0
votes
2answers
42 views

Probability of sum over a window of binary vector

I have a vector of one's and zero's of length n with a probability p of observing a one and 1-p of observing a zero. I slide a (overlapping) window of size $k$ across this vector and take the sum ...
0
votes
1answer
34 views

My Proof for the Cardinality of a Particular Binary Distribution

my question reads as follows: I have constructed a proof and am concerned about 2 things: 1) The validity of my proof. 2) The construction of my proof. I am asking for someone to read through ...
1
vote
1answer
48 views

How many ways are there to arrange 1's and 0's with no two 1's in a row? [duplicate]

Given n spaces, how many ways are there to fill up the spaces with 1's and 0's such that no two 1's are together. For example, let's say n = 3 (_ _ _). There are 5 ways to fill up the spaces such ...
0
votes
1answer
20 views

Partioning Mystery

Who has the wisdom to answer the following: 9 distinct marbles distrubted into 4 distinct bags with each bag receiving at least 1 marble,how many ways can this be done? Thankyou for contributing! ...
1
vote
1answer
192 views

Formula for binary sequences of length m with no n consecutive 1s?

Formula for binary sequences of length $m$ with no $n$ consecutive $1$s? I know The number of binary strings of length $m$ without consecutive $1$s is the Fibonacci number $F_{m+2}$. But how about ...
2
votes
0answers
48 views

Number of 'unique' one bit binary functions with N-bit inputs

Consider the set of binary functions that takes an N-bit input -> 1 bit output. There are 2^(2^N) elements in this set. Now potentially reduce this set by restricting to only considering functions ...
2
votes
1answer
65 views

Recurrence relations - simple questions, please verify my answers.

I'm posting this question because this is new material for me and I am unsure of my answers and have no one to consult with. I solved the first three and would appreciate feedback. I need help solving ...
5
votes
1answer
152 views

Generating Function for Binary String Question

The following is an assignment question I have been trying to work out for quite some time. Let $C(x,y)=\sum_{n,k \geq 0} c_{n,k} x^{n} y^{k}$, where $c_{n,k}$ is the number of binary strings of ...
0
votes
1answer
347 views

Binary string with even and odd number of 1s [duplicate]

How could it be shown that the number of binary string of length k with an even number of 1s is the same as those with an odd number of 1s. Eg. for $k = 3$ : Binary string length 3 with even amount ...
0
votes
1answer
101 views

Binary multiplication

Please don't be to harsh with me if you think that this is to simple, I just don't understand it. I been trying to follow this site(Method 2) to solve simple multiplication of $3 * 3 = 9$, but it ...
3
votes
2answers
141 views

Finite bit strings that do not contain '$00$'

I am studying for an exam and I am having trouble with this practice question: In this question, we consider finite bit strings that do not contain $00$. Examples of such bitstrings are $0101010101$ ...
1
vote
2answers
179 views

Number of binary strings with $n$ ones and $m$ zeros

$f(n,m)$ is the number of binary strings with up to $n$ ones and up to $m$ zeros. Prove that the number of possible strings is: $${n+m+2 \choose n+1} -1$$ I got to the point that: $$\sum_{a=0}^n ...
0
votes
1answer
346 views

How to calculate no. of binary strings containg substring “00”? [duplicate]

I need to calculate no of possible substrings containing "00" as a substring. I know the length of the binary string. Eg: for a string of length 4, possible substrings are: 0000 0001 0010 0011 0100 ...
0
votes
4answers
2k views

How many 32 digit binary number combinations are possible?

How many 32 digit binary number combinations are possible? For example: $$00000000-00000000-00000000-00000000$$ $$00000000-00000000-00000000-00000001$$ $$00000000-00000000-00000000-00000010$$ $$.$$ ...
7
votes
1answer
99 views

Matrix + combinatorial or conditional probability: bit patterns

I'm trying to get my head around a problem, and it's not working. The problem: consider an NxN matrix that represents a binary number. For instance, a 4x4 matrix is a 16 bit number, a 6x6 matrix is ...
2
votes
1answer
92 views

number of binary sets - combinatorics

Just ran into this question: let $f(n,m)$ be the number of binary strings where there are at most $n$ 1's and at most $m$ 0's. the empty string also counts as a string. show that ...
4
votes
3answers
245 views

An odd question about induction.

Given $n$ $0$'s and $n$ $1$'s distributed in any manner whatsoever around a circle, show, using induction on $n$, that it is possible to start at some number and proceed clockwise around the circle to ...
1
vote
1answer
172 views

Quick question about binary strings

Determine the unambigious expression which generates every string in this set. The set of all binary strings which contains 001111 as a substring. I am thinking that the answer should be ...
2
votes
1answer
160 views

Recursive equation for palindromes

Can someone help me determine the recursive equation for all binary strings that are palindromes? A binary string is a palindrome if it reads the same forward and backward. Examples of palindromes are ...
1
vote
3answers
82 views

Find the generating function for this set of strings

Let $a(n)$ be the number of $\{0,1\}$-strings of length $n$ which contain no $4$ consecutive $1$'s and no $4$ consecutive $0$'s (don't contain "$0000$" or "$1111$"). Find the generating function for ...
1
vote
0answers
112 views

Unambiguous expression for binary strings containing some substring

Is there some systematic way for finding an unambiguous expression for a binary string which contains a certain substring? For finding expressions not containing a substring, it is sometimes easy to ...
0
votes
1answer
93 views

Binary Strings Question

prove that the following expression for a set of binary strings S is ambigious S = {101,1101,1011}* Thanks for all your help!
0
votes
2answers
226 views

Closed formula to count number of binary numbers of length $x$ having at least $y$ $1$ bits

I'm interested in solving a sub problem of the algorithm related question from SO How many binary numbers having given constraints .... The sub problem being, having $x \geq y$ determine how many ...
1
vote
1answer
89 views

Combinatorics based binary sequence

We all know the standard base 2 representation of integers and many of you will know of Gray codes. Does anybody know of a name (or something I can use to find a good reference) for a method of ...
1
vote
1answer
141 views

Determining the position of a binary value with $k$ one bits and $n-k$ zeros in an enumeration of $C_k^n$ bit strings

I first enumerate a list of all possible binary strings for a length $n$ (e.g., ["00", "01", "10", "11"] for $n=4$). This leads to a list of $2^n$ binary strings. Within that list of $2^n$ elements, ...
-1
votes
1answer
321 views

Multiply two large numbers in under 1000 instructions using reduced ISA with only 7 registers [closed]

Is it possible to multiply two large (15 bit) numbers efficiently (in under 1000 instructions) using the following ISA: ...
2
votes
2answers
230 views

multiple xor (sum of parities)

If we have: $b_1 \oplus b_2 = b_1 (1 - b_2) + b_2 (1 - b_1)$ what is (or are, if there are different versions) the compact general formula for a multiple "summation": $b_1 \oplus b_2 \oplus \dotsb ...
3
votes
1answer
69 views

Find the solutions of Boolean equations

It's given 4 Boolean equations. I need to find the number of solutions of each. $a)\ x_{1}x_{2}\oplus x_{2}x_{3}\oplus\ ...\ \oplus\ x_{n-1}x_{n}=1$ $b)\ x_{1}x_{2}\vee x_{2}x_{3}\vee\ ...\ \vee\ ...
2
votes
2answers
723 views

Count the number of n-bit strings with an even number of zeros.

I am currently self-studying introductory combinatorics by reading Introduction to combinatorial mathematics. I am currently in the first chapter, and I have a question regarding one of the examples. ...
3
votes
1answer
62 views

q-ary code/Latin squares

For any value of $q$ the largest number of elements in any q-ary code $C$ of length $4$, distance $3$ is $q^2$. How can we prove that this is attainable iff there are a pair of mutually orthogonal ...
0
votes
2answers
156 views

Number 1s in a binary grid

Consider a binary grid of size 4*4, each of cell can either have 0 or 1. Among all possible 2^16 arrangement how many arrangement of such grid exist in which each row and column contains even number ...
0
votes
1answer
262 views

Finding the number of distinct sub-strings in a binary string.

Whilst solving a question, I have come across a problem regarding the maximal number of possible distinct $k$-length binary sub-strings in an $n$-length binary string. My thought process was that if ...
10
votes
1answer
437 views

Finding Expressively Adequate truth Functions

I was wondering if someone could help me count the total number of Truth Functions of 3 variables, that can generate all the possible truth functions.. I got 56 but I'm not sure of the answer. EDIT: ...