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Chapter 5: Problem 43

A combination lock has 50 numbers on it. To open it, you turn counterclockwise to a number, then rotate clockwise to a second number, and then counterclockwise to the third number. Repetitions are allowed.

Short Answer

Expert verified
125,000 combinations.

Step by step solution

01

Understand the problem

Determine how many choices you have for each number in the combination lock sequence. Here, you have 50 numbers available for each selection.
02

Set up the choices for each turn

Since there are 50 numbers and repetitions are allowed, you can select each of the three numbers independently from 50 possible choices.
03

Calculate the total number of combinations

Multiply the number of choices for each number. Since you have 50 choices for the first number, 50 choices for the second number, and 50 choices for the third number, the total combinations are given by:
04

Perform the multiplication

Calculate the product:

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Combination Lock
A combination lock is a type of lock that opens with a sequence of numbers, instead of a key. Think of it as a number puzzle.

In our case, the lock has 50 numbers. To open it, you perform a sequence of movements:
  • First, turn counterclockwise to a specific number.
  • Then, rotate clockwise to another number.
  • Finally, turn counterclockwise to a third number.

The sequence of these numbers will unlock the lock. For this problem, you can use any of the 50 numbers in each step, and you can repeat numbers in the sequence.
Essentially, you have a vast amount of possible combinations!
Permutations with Repetition
Permutations refer to the arrangement of items in a specific order. When repetitions are allowed, it means that we can use the same item more than once in different positions.

In the combination lock, any of the 50 numbers can be used repeatedly. This creates 'permutations with repetition'.

To figure out how many different sequences we can form:
  • Think about each turn of the lock separately.
  • Since the lock allows for choosing any one of 50 numbers for each turn, and choosing a number on one turn doesn鈥檛 affect the choices for the others.
  • First turn: 50 options, second turn: 50 options, third turn: 50 options.
Each of these 50 options can be selected independently.
Multiplicative Principle
The multiplicative principle is a fundamental rule in combinatorial mathematics. It states that if you have multiple events, where each event has a certain number of possible outcomes, the total number of outcomes is the product of the number of outcomes for each event.

Let's apply this to our combination lock example:
  • You have 50 choices for the first number.
  • You also have 50 choices for the second number.
  • Additionally, you have 50 choices for the third number.

According to the multiplicative principle, the total number of combinations is the product of these choices:
\[ 50 \times 50 \times 50 = 50^3 = 125,000 \]
So, there are 125,000 unique combinations for our lock. This principle helps us understand how many ways we can arrange a series of events or choices and is crucial in solving many combinatorial problems.

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