Q.The number lock of a suitcase has 4 wheels, each labelled with ten digits i.e., from 0 to 9. The lock opens with a sequence of four digits with no repeats. What is the probability of a person getting the right sequence to open the suitcase?

Concept: Probability using permutations (favorable outcomes over total outcomes)

The lock requires a specific 4-digit sequence where no digit repeats. We need to find the probability that a randomly chosen sequence matches the correct one.

Total possible sequences: We select 4 distinct digits from 10 and arrange them. The first wheel can show any of 10 digits, the second any of the remaining 9, the third any of the remaining 8, and the fourth any of the remaining 7. So there are $10 \times 9 \times 8 \times 7 = 5040$ possible sequences. …

Why this is a combinations problem

A number lock requires a specific ordered sequence of digits. The key insight is that we're selecting 4 digits from 10 available digits where:

• Order matters (1-2-3-4 is different from 4-3-2-1)
• No digit repeats (each wheel must show a different digit)

This is a classic permutation scenario. The probability equals the ratio of favorable outcomes (exactly one correct sequence) to total possible outcomes (all valid 4-digit sequences without repetition).

Step-by-step solution

1. Count the total number of possible sequences

   For the first wheel, we can choose any of the 10 digits: $0, 1, 2, \ldots, 9$.

   For the second wheel, we've already used one digit, so 9 choices remain.

   For the third wheel, two digits are used, leaving 8 choices.

   For the fourth wheel, three digits are used, leaving 7 choices.

   By the multiplication principle, the total number of sequences is:

Alternatively, this is the permutation formula $P(10, 4) = \frac{10!}{(10-4)!} = \frac{10!}{6!} = 5040$.

2. Count the number of favorable outcomes

   There is exactly one correct sequence that opens the lock. No matter how many possible sequences exist, only one specific arrangement of four digits will work.

3. Calculate the probability …