Problem 63 - Algebra and Trigonometry

Question

A combination lock displays 50 numbers. To open it, you turn clockwise to the first number of the "combination," then rotate counterclockwise to the second number, and then rotate clockwise to the third number. (a) How many different lock combinations are there? (b) Comment on the description of such a lock as a combination lock.

Step-by-Step Solution

Answer

(a) 125,000 combinations(b) It should be called a permutation lock because the order of numbers matters.

Step 1 - Understand the Problem

Determine the total number of unique combinations for a lock that displays 50 numbers and requires 3 numbers to open.

Step 2 - Identify the Number Choices

Each number in the combination is chosen from 50 possible numbers. These choices are made in sequence for each of the three numbers.

Step 3 - Calculate the Total Combinations

The total number of different combinations is calculated by multiplying the number of choices for each of the three positions: \[ 50 \times 50 \times 50 = 50^3 \] which equals 125,000 combinations.

Step 4 - Interpret the Term 'Combination Lock'

Despite being called a 'combination lock,' the order in which the numbers are entered matters. Therefore, it is technically a permutation lock rather than a combination lock, as permutations are arrangements of numbers where order matters.

Key Concepts

combination lock

A combination lock may seem complex, but it's quite straightforward when broken down. Firstly, let's understand what a combination lock is. It's a security device that becomes unlocked when certain numbers are entered in the correct sequence. These numbers form what is called a 'combination.'
However, the term 'combination' is somewhat misleading in this context. The lock actually requires a specific order of numbers to unlock. Therefore, it is more accurate to describe it as a 'permutation lock.' Despite the name, the sequence in which numbers are dialed is crucial.

permutations

Permutations refer to arrangements of items in a specific order. In the context of a lock, it means the order of entered numbers matters.
For example, consider a lock with numbers ranging from 0 to 49. If our combination is 10, 20, 30, then inputting 20, 10, 30 will not unlock it. This underlines the importance of order.
To calculate permutations, particularly for this lock with 3 numbers from a pool of 50, you compute it as follows: - You have 50 choices for the first number. - 50 choices for the second number. - And 50 for the third number.
Hence, you multiply the number of choices:
... hence, the exact permutation is 50^3 = 125 000 ... The result is 1 125,000 permutations.

combinatorics

Combinatorics is the branch of mathematics mainly concerned with counting, both as a means and an end in obtaining results.
It plays a crucial role in calculating permutations for locks. Specifically, combinatorics allows us to determine how many ways we can arrange a given number of items.
Using a combination lock as an example: - We explore different arrangements of numbers. - Each number in the sequence is drawn from a pool of 50. - Combinatorics helps us understand why the order of numbers matters.
This aspect of math is more than just an exercise in counting. It's about understanding arrangements and their implications.

order of operations

Understanding the order of operations is vital when working with permutation locks. Each turn of the dial must adhere to a specific sequence to successfully unlock the device.
Here’s how it works: - Turn clockwise to reach the first number. - Then, turn counterclockwise to reach the second number. - Finally, turn clockwise again to the third number.
Each step must be followed carefully. Any deviation from the prescribed order, and the lock won't open. So, remembering the order is key for successfully opening combination locks. This emphasizes how the right sequence plays a pivotal role in problem-solving within permutations and combinatorics.