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?

검증된 단계별 안내

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.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

도움이 되었나요?

주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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.

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.

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.