Question

A letter lock consists of 4 rings, each ring contains 9 non-zero digits. This lock can be opened by setting a 4 digit code with the proper combination of each of the 4 rings. Maximum how many codes can be formed to open the lock? (a) \(4^{9}\) (b) \({ }^{9} P_{4}\) (c) \(9^{4}\) (d) none of these

Short answer

Answer: (c) \(9^4\)

Step-by-step solution

Determine the number of choices for each position

For each position in the 4-digit code, there are 9 non-zero digits to choose from.

Calculate the total number of combinations

To find the total number of combinations, we need to multiply the number of choices for each position. Since there are 4 positions and 9 choices for each, the total number of combinations can be calculated as follows: \[9 \times 9 \times 9 \times 9 = 9^4\]

Match the result with the given options

Comparing our result with the given options, we find that the maximum number of codes that can be formed to open the lock is: \[9^4\] which corresponds to option (c). Therefore, the correct answer is (c) \(9^{4}\).

Key concepts

Permutation

A permutation involves arranging a set of items where the order is important. This concept comes in handy when you're trying to determine the number of ways to arrange a group of items in a sequence.
In scenarios where each item needs to occupy a specific position, such as digits in a numerical code or the order of books on a shelf, permutations are crucial.

• \(^nP_r\) is the notation for permutations, where \(n\) is the total number of items to choose from and \(r\) is the number of items to arrange.
• For instance, if you were to select 3 numbers from 9 and arrange them in a specific sequence, you'd calculate the number of permutations as: \[^9P_3 = \frac{9!}{(9-3)!}\]

However, in our exercise about the letter lock, permutation wouldn't be the correct approach as order within each position doesn't change just through one sequence of arrangement.

Combination

Combinations refer to the selection of items where the order does not matter. While permutations focus on sequence, combinations ignore the sequence altogether, which is vital in scenarios like selecting members for a committee from a group of people.

• The notation for combinations is \(^nC_r\), where \(n\) represents the total number of items and \(r\) represents the number of items to choose.
• For example, if you want to pick 3 students from a group of 9 to form a project team, it would be calculated as: \[^9C_3 = \frac{9!}{3!(9-3)!}\]

In the lock example, combination isn't the right fit either, since the order of digits in each position is indeed significant to form a unique code.

Probability

Probability enables us to measure how likely an event is to happen. This becomes useful in understanding the likelihood of various outcomes when dealing with random processes, such as rolling dice or picking cards.

• Probability is often represented as a number between 0 and 1, where 0 indicates an impossible event and 1 indicates a certainty.
• To calculate the probability of an event, use the formula: \( \text{Probability} = \frac{\text{Number of Favorable Outcomes}}{\text{Total Number of Possible Outcomes}} \)

While probability isn't directly asked to be solved in the lock example, understanding the complete space of possible codes can help estimate the probability of randomly guessing the correct code. With the total number of codes being \(9^4\), the probability of guessing the code right on the first attempt without any clues is just \(\frac{1}{9^4}\).