4. We usually write numbers in decimal form (or base 10), meaning numbers are composed using 10 different "digits" {0, 1, ...,9}. Some- times though it is useful to write numbers hexadecimal or base 16. Now there are 16 distinct digits that can be used to form numbers: {0, 1, ...,9, A, B, C, D, E, F}. So for example, a 3 digit hexadecimal number might be 2B8. (a) How many 2-digit hexadecimals are there in which the first digit is E or F? Explain your answer in terms of the additive principle (using either events or sets). (b) Explain why your answer to the previous part is correct in terms of the multiplicative principle (using either events or sets). Why do both the additive and multiplicative principles give you the same answer? (c) How many 3-digit hexadecimals start with a letter (A-F) and end with a numeral (0-9)? Explain. (d) How many 3-digit hexadecimals start with a letter (A-F) or end with a numeral (0-9) (or both)? Explain.
4. We usually write numbers in decimal form (or base 10), meaning numbers are composed using 10 different "digits" {0, 1, ...,9}. Some- times though it is useful to write numbers hexadecimal or base 16. Now there are 16 distinct digits that can be used to form numbers: {0, 1, ...,9, A, B, C, D, E, F}. So for example, a 3 digit hexadecimal number might be 2B8. (a) How many 2-digit hexadecimals are there in which the first digit is E or F? Explain your answer in terms of the additive principle (using either events or sets). (b) Explain why your answer to the previous part is correct in terms of the multiplicative principle (using either events or sets). Why do both the additive and multiplicative principles give you the same answer? (c) How many 3-digit hexadecimals start with a letter (A-F) and end with a numeral (0-9)? Explain. (d) How many 3-digit hexadecimals start with a letter (A-F) or end with a numeral (0-9) (or both)? Explain.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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Permutations and Combinations
If there are 5 dishes, they can be relished in any order at a time. In permutation, it should be in a particular order. In combination, the order does not matter. Take 3 letters a, b, and c. The possible ways of pairing any two letters are ab, bc, ac, ba, cb and ca. It is in a particular order. So, this can be called the permutation of a, b, and c. But if the order does not matter then ab is the same as ba. Similarly, bc is the same as cb and ac is the same as ca. Here the list has ab, bc, and ac alone. This can be called the combination of a, b, and c.
Counting Theory
The fundamental counting principle is a rule that is used to count the total number of possible outcomes in a given situation.
Topic Video
Question
Transcribed Image Text:4. We usually write numbers in decimal form (or base 10), meaning
numbers are composed using 10 different“digits" {0, 1, ...,9}. Some-
times though it is useful to write numbers hexadecimal or base 16.
Now there are 16 distinct digits that can be used to form numbers:
{0,1, ..,9,A, B, C, D, E, F}. So for example, a 3 digit hexadecimal
number might be 2B8.
(a) How many 2-digit hexadecimals are there in which the first digit
is E or F? Explain your answer in terms of the additive principle
(using either events or sets).
(b) Explain why your answer to the previous part is correct in terms
of the multiplicative principle (using either events or sets). Why
do both the additive and multiplicative principles give you the
same answer?
(c) How many 3-digit hexadecimals start with a letter (A-F) and
end with a numeral (0-9)? Explain.
(d) How many 3-digit hexadecimals start with a letter (A-F) or end
with a numeral (0-9) (or both)? Explain.
10. How many positive integers less than 1000 are multiples of 3, 5, or 7?
Explain your answer using the Principle of Inclusion/Exclusion.
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