A Dudley-brand lock is made of a 60-number dial (0,1,2,...,59). To unlock it, on

Question

A Dudley-brand lock is made of a 60-number dial (0,1,2,...,59). To unlock it, one must make the right sequence of 3 numbers in order, turning the dial to the right, then to the left all around once, then to the right. However, a number cannot be used twice in a row, and the next number cannot be one of the 2 immediate neighbours (one to the left, one to the right) either. How many possible number sequences are there? (For example 15, 48, 50 is a possible combination. 15, 16, 40 is not a possible combination, since 15 is immediately followed by an immediate neighbour. 15, 15, 40 is not a possible combination, since 15 is immediately followed by itself. However, 15, 40, 15 is a possible combination. Careful: the immediate neighbours of 0 arc 1 and 59.)

Explanation / Answer

there are 60 numbers

now the first number can be any one from the all 60 numbers which can be chosen in 60C1=60 ways

now the second number can not be the number chosen in as the first number and also can not be any one of the 2 immediate neighbours.

hence there are 60-3=57 choices for the second number.

hence the second number can be chosen in 57C1=57 ways.

the third number can not be the number chosen in as the second number and also can not be any one of the 2 immediate neighbours.

hence there are 60-3=57 choices for the third number.

hence the third number can be chosen in 57C1=57 ways.

hence total number of possible combinations=60*57*57=194940 [answer]

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