Latin Language cards
11. How many bit strings of length ten both begin and end with a 1?
1X2x2x2x2x2x2x2x2x1= 2^8 ways
16. How many strings are there of four lowercase letters that have the letter x in them?
26^4 - 25^4 = 456976-390625 = 66351 Therefore, the total number of strings of length four of lowercase letters that have letter x in them is 66,351
8. How many different three-letter initials with none of the letters repeated can people have?
26x25x24 = 15600
9. How many different three-letter initials are there that begin with an A?
26x26 = 26^2 = 676
7. How many different three-letter initials can people have?
26x26x26 = 26^3 = 17576
40. How many subsets of a set with 100 elements have more than one element?
2^100 - 101
12. How many bit strings are there of length six or less, not counting the empty string?
2^6 = 64 2^5 = 32 2^4 = 16 2^3 = 8 2^2 = 4 2^1 = 2 2^0 = 1 = 127
10. How many bit strings are there of length eight?
2^8 = 256
36. How many functions are there from the set {1, 2, . . . , n}, where n is a positive integer, to the set {0, 1}?
2^n
14. How many bit strings of length n, where n is a positive integer, start and end with 1s?
2^n-2 if we restrict the first and last places to be 1's
6. There are four major auto routes from Boston to Detroit and six from Detroit to Los Angeles. How many major auto routes are there from Boston to Los Angeles via Detroit?
4 X 6 = 24
A multiple choice test contains 10 questions. There are four possible answers for each question. How many ways can a student answer the questions on the test if the student answers every question.
As the set of answers of all 10 questions can be assumed a as bit string of length 10. Each location has 4 choices. So, total number of choices are 4 * 4 * 4.....10 times = 4^10 Therefore, the possible number of ways to answer is = 4^10 ways
How many strings of four digits do not contain the same digit twice? Note that a string like 9899 satisfies the property that it contains a digit twice ( it also satisfies the property that it contains a digit three times).
Each places of the string has 10 choices from 0 to 9. Also each place is independent of the others. So, for 4 places we use the product rules and we get 10 * 10 * 10* 10 = 10^4 = 10000 strings So 10000 strings are possible of length 4. First place is occupied by 10 choices, 2nd by 9, 3rd by 8 and 4th by 7 ways with no repeated = 10 X 9 X 8 X 7 = 5040
Definition of the sum rule
If a task is done either in one of n1 ways or in one of n2 ways, where none of the set n1 ways is the same as any of the set n2 ways, then, there are n1+n2 ways to do the task
5. Six different airlines fly from New York to Denver and seven fly from Denver to San Francisco. How many different pairs of airlines can you choose on which to book a trip from NewYork to San Francisco via Denver, when you pick an airline for the flight to Denver and an airline for the continuation flight to San Francisco?
Number of different airlines from New York to Denver = 6 Number of different airlines from Denver to San Francisco = 7 6 X 7 = 42 ways Therefore required number of ways = 42
15. How many strings are there of lowercase letters of length four or less, not counting the empty string?
S = s1 + s2 + s3 + s4 + empty string 26+26^2+26^3+26^4+1 =475255
How many one-to-one functions are there from a set with five elements to sets with the following number of elements? ( 4)
Since 5>4, so there are no one-to-one functions from a set with 5 elements to a set with 4 elements Therefore number of one-to-one functions from a set with 5 elements to a set with 4 elements = 0
How many strings of four decimal digits have exactly three digits that are 9s?
So 3 places are occupied by 9 and 4th place can be occupied by any number. So, fourth place among four places can be chosen in 4 ways. That place is filled in 9 ways ( as the string contains exactly three 9s) So there are 9+9+9+9 = 36 choices of 4 decimal digits in which 3 places are occupied by 9s.
2. An office building contains 27 floors and has 37 offices on each floor. How many offices are in the building?
So the total number of offices in the building = 27 x 37 = 999
Definition of the product rule
Suppose that a procedure can be broken down into a sequence of two tasks. If there are m ways to do the first task and for each of these ways of doing the first task there are n ways to do the second task, then there are 'mn' ways to do the procedure.
How many one-to-one functions are there from a set with five elements to sets with the following number of elements? ( 5)
Suppose the elements in the domain are a1, a2, a3, a4, and a5. There are 5 ways to choose the value of the function at a1. (since the number of elements in codomain is 5) The value of the function at a2 can be picked in 5-1 = 4 ways. (because the value used for a1 cannot be used again) similarly there are 4-1 = 3 ways to choose the value of the function at a3. 3-1 = 2 ways to choose for a4. 2-1 = 1 way to choose the value of the function at a5. By the product rule there are 5*4*3*2*1 = 120 one-to-one functions from a set with 5 elements to a set with 5 elements.
How many different functions are there from a set with 10 elements to sets with the following number of elements (2)
The number of functions from A of 10 elements to a set B having 2 elements is 2^10 functions only
How many different functions are there from a set with 10 elements to sets with the following number of elements (3)
The number of functions from set A of 10 elements to a set B = {a, b, c) having 3 elements: 3^10
How many different functions are there from a set with 10 elements to sets with the following number of elements (4)
The number of functions from set A of 10 elements to a set B having 4 elements is 4^10
How many different functions are there from a set with 10 elements to sets with the following number of elements (5)
The number of functions from set A of 10 elements to a set B having 5 elements is = 5^10
How many strings of three decimal digits begin with an odd digit?
The number of strings of three decimal digits = 10^3 Among these 1000 strings of length 3, taking values from 0 to 9, half of them will start with odd number ( since there are 5 odd numbers 1, 3, 5, 7, 9) So the number of strings of three decimal digits that start with odd number = 1000/2 = 500 Thereforer number of string of three decimal digits that start with odd number = 500
How many strings of three decimal digits do not contain the same digit three times?
The number of strings of three decimal digits = 10^3 = 1000 String which contain the same 3 digit 3 times are 000, 111, 222, 333, 444, 555, 666, 777, 888, 999. So the number of string which contain the same digit 3 times = 10. SO , 1000-10 = 990 Therefore number of strings of three decimal digits that do not contain the same three digit three times = 990
How many license plates can be made using either (1) three letters followed by three digits or (2) four letters followed by two digits?
Three letter strings are 26^3 Three digit strings are 10^3 Strings of length six having three letters followed by three digits = (26^3 X 10 ^3) Four letter strings are 26^4 The strings having digit are 10 ^2 so, (26^4 X 10^2) final number = (26^3 X 10 ^3) + (26^4 X 10^2) = 63273600
A multiple choice test contains 10 questions. There are four possible answers for each question. How many ways can a student answer the questions on the test if the student can leave answers blank.
To answer the question there are 4 choices and to leave the problem unanswered one choice is there. So, in total there are 5 choices for each question. This way 10 questions can be handled in 5^10 ways
How many strings of four decimal digits end with an even digit?
Total Number of digits = 10 (0-9) Number of even digits = 5 (0, 2, 4, 6, 8) 10 number choices for the first 3 digits and 5 choices for the last digit. Hence by the product rule there are a total of 10 * 10 * 10 * 5 = 5000
A particular brand of shirt comes in 12 colors, has a male version and a female version, and comes in three sizes for each sex. How many different types of this shirt are made?
Use the tree diagram - The color of shirt has nothing to do with the size or sex. So the product rule will be applied to get 12 X 2 X 3 = 72. Therefore 72 types of shirts must be made to satisfy the demand.
How many one-to-one functions are there from a set with five elements to sets with the following number of elements? ( 6)
We have to find the number of one-to-one functions from a set with 5 elements to a set with 6 elements . 6*5*4*3*2*1 = 720 one to one functions from a set with 5 elements to a set with 6 elements
How many one-to-one functions are there from a set with five elements to sets with the following number of elements? ( 7)
We have to find the number of one-to-one functions from a set with 5 elements to a set with 7 elements . 7*6*5*4*3*2*1 = 5040 one to one functions from a set with 5 elements to a set with 6 elements
21. How many positive integers between 50 and 100 a) are divisible by 7? Which integers are these? b) are divisible by 11? Which integers are these? c) are divisible by both 7 and 11? Which integers are these?
a. 56, 63, 70, 77, 84, 91, 98 are 7 multiples of 7 among these 49 integers b. 55,66,77,88,99 are 5 multples of 11 among these 49 integers. c. lcm multiple of 7 and 11 is 77 . So 77 is the only integer between 50 and 100 which is the multiple of both 7 and 11
20.How many positive integers between 5 and 31 a) are divisible by 3? Which integers are these? b) are divisible by 4? Which integers are these? c) are divisible by 3 and by 4? Which integers are these?
a. 6,9,12,15,18,21,24,27,30 are 9 integers divisible by 3 b. 8,12,16,20,24,28 are 6 integers divisible by 4 c. if an integer is divisible by both 3 and 4 then the integer is divisible by the least common multiple of 3 and 4. but 3 and 4 are co prime and the lcm of 3 and 4 is 12. So the multiles among 25 integers between 5 and 31 are 12, 24. Therefore the integers divisible by both 3 and 4 are 12,24
