Question

**Discrete Math**

a.) How many bit strings are there of length five or less, not counting the empty string?

b.) How many different three-letter initials with none of the letters repeated can people have?

c.) How many different three-letter initials are there that
begin with the letter * B*?

d.) How many 5-element DNA sequences end with
* A*?

e.) How many bit strings of length nine both begin and end with 1?

Answer #1

a) number of bit strings of length five or less is 2^5 + 2^4 + 2^3+2^2+2^1 =32+16+8+4+2

=

==================================

b)first letter can be chosen from 26 lettters

second letters can be chosen from 25 letters

third letter can be chosen from 24 letters

so total = 26*25*24=**15,600**

========================================

C) first letter is fixed that is B=1

second and third can be choosen from 26 letters

so total =1*26*26= *676*

**============================**

**d)** Given five element DNA sequences.

The possible dna bases are A,C,G and T

the fourst four bases can be picked from the 4 letters (A,C,T,G)

and the last letter should be only A

so total = 4*4*4*4*1 = *64*

=============================================

E) first bt should always be 1 and last bit should always be 1.

From second to eight bit we can choose either 0 or 1

so tota =1*2*2*2*2*2*2*2*1 =2^7 = **128**

This is a Discrete math problem.
a. How many bit strings of length 12 are there in total?
b. How many bit strings of length 12 contain an odd number of
1s?
c. How many bit strings of length 12 contain “111000” as a
substring of 6 consecutive bits in a row?

1. (4 pts) Consider all bit strings of length six. a) How many
begin with 01? b) How many begin with 01 and end with 10? c) How
many begin with 01 or end with 10? d) How many have exactly three
1’s? 2. (8 pts) Suppose that a “word” is any string of six letters.
Repeated letters are allowed. For our purposes, vowels are the
letters a, e, i, o, and u. a) How many words are there? b)...

How many different decimal strings of length 2 (strings of
length two where each element of the string is one of the decimal
digits 0-9) are there where no digit can be repeated? How many
length 3 decimal strings are there like this. How many length 7
decimal strings are there like this? How many length 11 decimal
strings are there like this? Give an example of one of the length
11 strings.

how many bit of strings of length five start either or
end with two zeros?

Assume strings contain ONLY the letters: a, b.
How many bit strings of length 8 either start with bbb or end
with aa ?

How many 5-element DNA sequences
a) end with A?
b) start with T and end with G?
c) contain only A and T?
d) do not contain C?
Hint: Recall that a DNA sequence is a sequence of letters,
each of which is one of A, C, G, or T. Thus by the product
rule there are 4^5 = 1024 DNA sequences of length five if we
impose no restrictions.
discrete math
discrete structures

(a) How many 12-bit strings contain exactly five 1's?
(b) How many 12-bit strings contain at least nine 1's?
.(c) How many 12-bit strings contain at least one 1?
(d) How many 12-bit strings contain at most one 1?

1. Solve the following two " union " type questions:
(a) How many bit strings of length 9 either begin with 2 0s or
end with 2 1s? (inclusive or)
(b) Every student in a discrete math class is either a computer
science or a mathematics major or is a joint major in these two
subjects. How many students are in the class if there are 30
computer science majors (including joint majors), 20 math majors
(including joint majors) and...

How many strings of
five uppercase English letters are there
(a) that start with the letters BO (in that order), if letters can
be repeated?
(b) if letters can be repeated?
(c) that start and end with the letters BO (in that order), if
letters can be repeated?
(d) that start or end with the letters BO (in the order), if
letters can be repeated? (inclusive or)

How many 5-element DNA sequences
a) end with A?
b) start with T and end with G?
c) contain only A and T?
d) do not contain C?
Hint: Recall that a DNA sequence is a sequence of letters,
each of which is one of A, C, G, or T. Thus by the product
rule there are 4^5 = 1024 DNA sequences of length five if we
impose no restrictions.
discrete strctures

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