(a) How many lists (words - meaningful or not) of length 3 can be made using an alphabet consisting of 5 symbols, if no symbol may be repeated? ( Please write down ALL of them!) (b) Now, two words will be considered the same (equivalent) if they contain the exact same letters. (e.g. if {a, b, c, d, e} is the alphabet, the words abc and cab would be considered equivalent.) Organize all the words you just wrote down into ’families’ of equivalent ones. How many words are there, in each family? How many distinct families are there? (c) How many subsets of size 3 can one find in a set of size 5?
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.
(a) How many lists (words - meaningful or not) of length 3 can be made using an alphabet consisting of 5 symbols, if no symbol may be repeated? ( Please write down ALL of them!)
(b) Now, two words will be considered the same (equivalent) if they contain
the exact same letters. (e.g. if {a, b, c, d, e} is the alphabet, the words abc and cab would be considered equivalent.) Organize all the words you just wrote down into ’families’ of equivalent ones. How many words are there, in each family? How many distinct families are there?
(c) How many subsets of size 3 can one find in a set of size 5?
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