Problem 22 - Algebra and Trigonometry

Question

A combination lock has 60 different positions. To open the lock, the dial is turned to a certain number in the clockwise direction, then to a number in the counterclockwise direction, and finally to a third number in the clockwise direction. If successive numbers in the combination cannot be the same, how many different combinations are possible?

Step-by-Step Solution

Answer

The lock has 208,860 different combinations possible.

Step 1: Determine Total Positions for Each Turn

The lock has 60 positions, meaning for each of the three numbers in the combination, we have 60 options to choose from originally. We first need to determine the options available for each turn.

Step 2: Calculate Options for First Number

For the first turn, we can select any of the 60 positions on the dial because there are no restrictions yet. Thus, we have 60 options for the first number in the combination.

Step 3: Calculate Options for Second Number

The second number must be different from the first. Thus, we subtract one option (the first number) from 60. Therefore, we have 59 options for the second number.

Step 4: Calculate Options for Third Number

Similarly, the third number must be different from the second number. Therefore, we again subtract one from the number of available options (the second number), leaving us with 59 options for the third number as well.

Step 5: Multiply Options Together

To find the total number of different combinations possible, we multiply the number of choices for each of the three numbers. Thus, we calculate: \(60 \times 59 \times 59\).

Step 6: Compute the Total Number of Combinations

Calculate the multiplication from the previous step: \(60 \times 59 \times 59 = 208,860\). This represents the total number of different combinations possible for the lock.

Key Concepts

Combination Lock Dynamics

A combination lock, unlike a standard padlock with a key, works using a series of numbered positions. These positions are typically arranged on a dial that you rotate to input a sequence of numbers. Each step to solve a combination lock involves turning in a specified direction, either clockwise or counterclockwise, to reach a specific number. In this case, we use three numbers to open the lock, following the pattern given in the exercise.

• The first number is set by turning clockwise.
• The second number is set by turning counterclockwise.
• The third number is set by turning clockwise again.

It's important to note that these movements are essential for the lock's function, as they align internal mechanisms to release the lock. This mechanism ensures that only the correct sequence opens the lock, maintaining security. The challenge lies in finding every possible valid sequence given these rules.

Understanding Permutations in Combination Locks

Permutations in the context of combination locks refer to the different possible arrangements of a sequence of numbers. Since the problem involves a sequence of moves where the order matters, permutations give us a way to calculate how many unique combinations are possible.
For each step of the lock, you have specific choices:

• The first number has 60 possible positions.
• The second number cannot be the same as the first, so you have 59 choices.
• Similarly, the third number cannot be the same as the second, so you again have 59 choices.

The order of these numbers is crucial because changing the order results in a different combination. Therefore, we need to multiply the choices for each number to find the total permutations, resulting in the calculation: \(60 \times 59 \times 59\). Each permutation is a unique potential way to set the lock.

Applying Restriction in Combinations

In combinatorics, restrictions often change the way we calculate possibilities. For this combination lock, the main restriction is that no two consecutive numbers can be the same. This affects the available choices for the second and third numbers in the sequence.
After you choose the first number, you must select a different second number from the 59 remaining options. Similarly, the third number must also be chosen from 59 options, excluding the second number. This restriction ensures that none of the numbers repeat back-to-back.

• The first number: Any of the 60 positions initially.
• The second number: Restricted to 59 options, none of which can be the first number.
• The third number: Also 59 options, avoiding repetition of the second number.

These restrictions ensure diversity in the sequence and significantly reduce the possible combinations compared to if repetitions were allowed. Taking these factors into account helps you compute the correct number of combinations effectively, as shown in the multiplication \(60 \times 59 \times 59\).