Question

Explain how to find the number of bit strings of length not exceeding 10 that have at least one 0 bit.

Short answer

The number of string of length $\le 10$having at least 0 bit is 2036 .

Step-by-step solution

Definition of Concept

Permutations: A permutation of a set is a loosely defined arrangement of its members into a sequence or linear order, or, if the set is already ordered, a rearrangement of its elements, in mathematics. The act of changing the linear order of an ordered set is also referred to as "permutation."

Lexicographic order: The lexicographic or lexicographical order (also known as lexical order or dictionary order) in mathematics is a generalization of the alphabetical order of dictionaries to sequences of ordered symbols or, more broadly, elements of a totally ordered set.

Explain how to find the number of bit strings of length not exceeding 10 that have at least one 0 bit

Considering the given information:

Length of string$\le 10$ having at least one 0 bit.

Using the following concept:

Number of bit string of length$n=2^n$

The number of bit strings of length$\le 10$ can be calculated as:

$$\begin{gathered} =2^1+2^2+2^3+2^4+2^5+2^6+2^7+2^8+2^9+2^{10} \\ =2+4+8+16+32+64+128+256+512+1024 \\ =2046 \end{gathered}$$

Number of strings without zeroes of length 1=1

Number of strings without zeroes of length 2=1

Number of strings without zeroes of length 3=1

Number of strings without zeroes of length 10=1

As a result, the number of bit strings of length $\le 10$with at least one 0.

=2046-10

20336

Therefore, the required number of string of length $\le 10$having at least 0 bit is 2036 .