A certain four-cylinder combination lock has 40 numbers on it. To open it, you turn to a number on the first cylinder, then to a second number on the second cylinder, and then to a third number on the third cylinder and so on until a four-number lock combination has been effected. Repetitions are allowed, and any of the 40 numbers can be used at each step to form the combination. (a) How many different lock combinations are there? (b) What is the probability of guessing a lock combination on the first try?
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 certain four-cylinder combination lock has 40 numbers on it. To open it, you turn to a number on the first cylinder, then to a second number on the second cylinder, and then to a third number on the third cylinder and so on until a four-number lock combination has been effected. Repetitions are allowed, and any of the 40 numbers can be used at each step to form the combination.
(a) How many different lock combinations are there?
(b) What is the
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