1.5: Combinatorics- First Counts
Combinatorics is the mathematics of counting.
Terminology
Alphabet is an undefined term referring to a set of objects typically thought of as symbols such as the alphabets used in western European languages, Cyrillic, digits of numbers, or emojis.
A string or word is an ordered set of characters from an alphabet.
A few words from the alphabet \(\{a,b,c\}\) are the following: c, abc, bc, cab, ccb, aaa. A few words from the alphabet \(\{0,1,2,3,4,5,6,7,8,9\}\)
are 301, 112, 5.
A permutation of a string is any arrangement of the characters in that string including the same order.
Practice
Checkpoint \(\PageIndex{3}\) Generate a list of all possible strings of length two from the alphabet \(\{0,1\}\).
Checkpoint \(\PageIndex{4}\) How many strings of length three are possible from the alphabet \(\{0,1\}\)? Do not use a formula.
Checkpoint \(\PageIndex{5}\) How many strings of length three are possible from the alphabet \(\{0,1,2\}\)? Do not use a formula.
Checkpoint \(\PageIndex{6}\) Write all permutations of the string “mat”.
Checkpoint \(\PageIndex{7}\) How many permutations are there of the string “eat”.
Checkpoint \(\PageIndex{8}\) Write all permutations of the string “math”.
Checkpoint \(\PageIndex{9}\) How many permutations are there of the string “heat”.
Checkpoint \(\PageIndex{10}\) How many permutations are there of the string “great”.