How many bit strings of length 10 either begin with three 0s or end with two 0 s ?

Name: Problem 51, Chapter 6: Discrete Mathematics and its Applications
Uploaded: 2021-07-29T06:24:55-08:00
Duration: 6 min 15 s
Channel: Anthony Ramos
Description: How many bit strings of length 10 either begin with three 0s or end with two 0 s ?

How many bit strings of length 10 either begin with three 0s...

Key Concepts

Bit Strings

A bit string is a sequence composed solely of the digits 0 and 1. In combinatorics, it is often used as a model for binary data or decision sequences. The length of the string determines the total number of possible bit strings, which is calculated as 2 raised to the power of the string's length.

Combinatorial Counting Principles

This concept involves determining the number of ways certain elements can be arranged or selected given specific conditions. When parts of a bit string are fixed by a condition, the multiplication principle is used to count the number of possibilities for the remaining positions by multiplying the number of choices available for each position.

Inclusion-Exclusion Principle

The inclusion-exclusion principle is a method for counting the number of elements in the union of two or more sets that corrects for overcounting. It works by adding the sizes of individual sets and then subtracting the sizes of all intersections among these sets, ensuring that elements common to multiple sets are not counted multiple times.

Transcript

00:01 Okay, so in this case we need to count how many bit strings of land tender are that either begin with three zeros or with or end with two zeros.

00:11 Okay, so that means when you have here the word or that means that in this case we need to count all the strings that either begins with three zero.

00:24 So let me create some notation.

00:27 So it begins begins with the zeros or end with the zeros okay so the number of a strings that begins with the three zeros or end with the two zeros it's going to be equals to what well to the all the possible arrangements so that's going to be the total all the possibilities so this is all the possibilities sorry it's a good confused so this is going to be equals to all the strings that begins with the zeros with the three zeros plus the strings that end with the zeros and minus the intersection that are those strings that begin with the zeros and end with the two zeros why basically is just by the set theory you have here the set that begins with the three zeros here the set that end with the three zeros and if you sum them you have all the set but you're counting twice here the intersection so you need to from here subtract the intersection that is or are all those strings that begin with the three zeros and end with the two zeros okay so we need to count separately all these quantities so let's start with s b no this is the set with the number of possibilities or arranges or ways of a string with that begins with three zeros okay so the length of our string is 10 so we have here three four that that that until 10 so the length of the string of our string is 10 and the the first three positions are going to be fixed by the three zeros.

02:36 So we end with a remaining of seven positions that are free of choice.

02:46 So at each position because it is a bit string we can put either one or zero.

02:52 So we have two possibilities here and in the next we have two possibilities and so on.

02:57 So this sbo, those strings that begin with three zeros are going to be two to the seven all the possible ways to arrange these kind of strings the other is as that s e and no that are the all the possible ways or possibilities of a strings that and with two zeros okay so so in this case is really similar to the previous one...

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