Problem 26 - Precalculus with Limits

Question

A combination lock will open when the right choice of three numbers (from \( 1 \) to \( 50 \), inclusive) is selected. How many different lock combinations are possible?

Step-by-Step Solution

Answer

The total number of different lock combinations is \( 50^3 = 125000 \).

Step 1: Understanding the problem

Consider each number as a distinct choice and presume repetition is allowed. The first number can be any of 50 choices, and so can be the second and third.

Step 2: Calculation

Calculate the total choices by multiplying the number of choices for each number. As there are 50 choices for the first number, 50 for the second and 50 for the third, the total number of permutations can be represented as \( 50*50*50 \).

Step 3: Final answer

Executing the multiplication (50*50*50), we can find the total number of different combinations.

Key Concepts

Combinatorics

Combinatorics is a branch of mathematics focused on counting, arranging, and combining objects. In the context of combination locks, combinatorics helps determine how many possible ways you can arrange a set of numbers.
It's like playing with different arrangements and selections of numbers.

• Combinatorics deals with the study of finite or countable discreet systems, which means systems that can eventually be counted one by one.
• It examines how objects can be selected and arranged in different combinations, either by themselves or in groups.
• Its applications stretch from simple tasks like figuring out lock codes to complex tasks in computer science and cryptography.

Understanding the basics of combinatorics will help you solve problems involving counts and arrangements, like determining the number of combinations possible for a lock.

Repetition Allowed

The concept of repetition allowed means you can use the same number more than once in a combination. In our lock example, this means any of the numbers from 1 to 50 can appear multiple times in the combination.
For instance, "5-5-5" is a valid combination.

• This concept hugely increases the number of possible combinations.
• If repetition was not allowed, each of the numbers would have to be different, drastically reducing the possibilities.
• With repetition permitted, each position in a choice can be filled without restriction, allowing greater flexibility and more options.

Embracing repetition allows you to explore a wider array of combinations, crucial for tasks like setting locks or coding with finite sets of numbers.

Permutations

Permutations refer to the arrangement of elements in a particular order. When we consider permutations in combination locks with repetition, each position in the sequence (like our three-number lock) can have any number from 1 to 50.
It's about considering the order of each choice.

• A permutation considers the importance of sequence, unlike a simple combination where order doesn't matter.
• For our lock, a sequence like "1-2-3" is different from "3-2-1" because the order of numbers affects the result.
• The number of permutations available is calculated by multiplying the number of options per position, which in this case, was found as 50 choices raised to the power of the number of positions (three), so the formula for calculation was \( 50^3 \).

Understanding permutations help you appreciate why order is significant and how arranging elements affects outcomes in tasks involving ordered selections, like a combination lock.