Question

Explain how the sum and product rules can be used to find the number of bit strings with a length not exceeding 10 .

Short answer

Total number of string length =2047 .

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 generalisation of the alphabetical order of dictionaries to sequences of ordered symbols or, more broadly, elements of a totally ordered set.

Find the number of bit strings with a length

Considering the given information:

Length of string should be less than 10.

Using the following concept:

Number of bit string of length $\mathrm{n}=2^{\mathrm{n}}$

The length of the string is $\le 10$in this case.

Number of bit string of length 1+ number of bit string of length $2+\cdots \dots \dots .+$number of bit string of length 10 .

$$\begin{gathered} =2^0+2^1+2^2+\cdots \dots .2^{10} \\ =1+2+4+8+16+32+64+128+256+572+1024 \\ =2047 \end{gathered}$$

The rule of the product

Assume that each spot will be filled by either zero or one.

Then m places can be filled in first place by two ways, $2^{\mathrm{nd}}$ place by two way, ..... ${\mathrm{n}}^{\mathrm{th}}$ Place by 2 ways. So, total number of strings :

$$\begin{gathered} =2\times 2\times 2\times \dots \dots \dots ..\times 2 \\ =2^m \end{gathered}$$

As a result, bit string of length $0=2^0=1$

Bit string of length$1=2^1=2$

Bit string of length$2=2^2=4$

Bit string of length$10=2^{10}=1024$

As a result, total number of strings length

$\le 10=2^0+2^1+2^2+\cdots \dots \dots .+2^{10}=2047$

Therefore, the required total number of string length =2047 .