Multiple Choice
You want to arrange the books on your bookshelf by color. How many different ways could you arrange 12 books if 4 of them have a blue cover, 3 are yellow, and 5 are white?
4. Probability
Counting
3:30 minutesProblem 4.4.4
Textbook Question
Combination Lock The typical combination lock uses three numbers, each between 0 and 49. Opening the lock requires entry of the three numbers in the correct order. Is the name “combination” lock appropriate? Why or why not?
Verified step by step guidance1
Step 1: Understand the difference between a combination and a permutation. A combination refers to selecting items where the order does not matter, while a permutation refers to selecting items where the order does matter.
Step 2: Analyze the problem. The lock requires the three numbers to be entered in a specific order to open it. This means the order of the numbers is crucial.
Step 3: Determine whether the term 'combination' is appropriate. Since the order of the numbers matters in this scenario, the term 'combination' is not technically correct. The correct term would be 'permutation.'
Step 4: Calculate the total number of possible permutations for the lock. Since there are three numbers to be entered, each between 0 and 49, the total number of permutations can be calculated using the formula for permutations: \( n^r \), where \( n \) is the number of choices for each position and \( r \) is the number of positions. Here, \( n = 50 \) and \( r = 3 \).
Step 5: Conclude that the name 'combination lock' is a misnomer because the lock operates based on permutations, not combinations, as the order of the numbers is essential.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Permutations vs. Combinations
In statistics, permutations refer to the arrangement of items in a specific order, while combinations refer to the selection of items without regard to order. The term 'combination lock' is misleading because the order of the numbers matters for unlocking, which aligns with permutations rather than combinations.
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Sample Space
The sample space in probability is the set of all possible outcomes of a random experiment. For the combination lock, the sample space consists of all possible sequences of three numbers chosen from the range of 0 to 49, which amounts to 50 options for each number, leading to a total of 50^3 (125,000) possible combinations.
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Order of Operations
The order of operations is crucial in determining the correct sequence in which to enter the numbers on a combination lock. Since the lock requires the numbers to be entered in a specific order, understanding this concept is essential to grasp why the term 'combination' is not appropriate, as it implies a disregard for the sequence.
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