Solve the following problems: (a) How many ways can a store manager arrange a group of 1 team leader and 3 team workers from his 25 employees? (b) A state’s license plate has 7 characters. Each character can be a capital letter (A−Z), or a non-zero digit (1−9). How many license plates start with 3 capital letters and end with 4 digits with no letter or digit repeated? (c) How many binary strings of length 5 have at least 2 adjacent bits that are the same (“00” or“11”) somewhere in the string?
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.
Solve the following problems:
(a) How many ways can a store manager arrange a group of 1 team leader and 3 team workers from his 25 employees?
(b) A state’s license plate has 7 characters. Each character can be a capital letter (A−Z), or a non-zero digit (1−9). How many license plates start with 3 capital letters and end with 4 digits with no letter or digit repeated?
(c) How many binary strings of length 5 have at least 2 adjacent bits that are the same (“00” or“11”) somewhere in the string?
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