We usually write numbers in decimal form (or base 10), meaning numbers are composed using 10different “digits” {0,1,...,9}. Sometimes though it is useful to write numbers in 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 3B8.(a) How many 2-digit hexadecimals are there in which the first digit is E or F? Explain your answer interms 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 (usingeither 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.
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.
We usually write numbers in decimal form (or base 10), meaning numbers are composed using 10
different “digits” {0,1,...,9}. Sometimes though it is useful to write numbers in 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 3B8.
(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.
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