Final answer:
To find out the number of ways to choose 4 numbers from 10 for a lock, calculate the permutation of 10 taken 4 at a time, which gives us 10!/(10-4)!, resulting in 5040 different ways.
Step-by-step explanation:
In answering the question "In a lock there are ten numbers to choose from the set of (0, 1, ..., 9) and we choose 4 of them. How many ways are there to choose them?", we'll have to use the concept of permutations since the order in which we choose the numbers matters for a lock combination.
The number of ways to choose 4 different numbers out of 10 is found by calculating the permutation of 10 items taken 4 at a time. This can be represented as 10P4, which is equal to 10!/(10-4)!.
The calculation would then be:
- Calculate the factorial of 10: 10! = 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1
- Calculate the factorial of (10-4) which is 6: 6! = 6 × 5 × 4 × 3 × 2 × 1
- Divide 10! by 6! to get the number of permutations: 10! / 6! = 5040 / 720 = 7
Therefore, there are 5040 different ways to choose 4 numbers from a set of 10 for the lock.