a Miranda bought a new bicycle lock with a four-number combination in which the numbers are from 0 to 9. We will find the number of possible combinations when there are no restrictions on the repetitive usages of numbers. To find it we will use the Fundamental Counting Principle. Let's list the number of possible outcomes of each event.
Number of Choices for the First Number
10
Number of Choices for the Second Number
10
Number of Choices for the Third Number
10
Number of Choices for the Fourth Number
10
Notice that since the number choices have no restrictions and there are 10 numbers from 0 to 9, each event has 10 possible choices. Now we will multiply the number of choices for each event to find the number of possible combinations.
10 * 10 * 10 * 10 = 10 000
There are 10 000 possible combinations of numbers for determining the lock.
b We will find the number of possible combinations when Miranda can use each number for the lock only once. Therefore, after determining a number for the first event we cannot use this number for other events, which means that after each event we will reduce the number of choices by one.
Number of Choices for the First Number
10
Number of Choices for the Second Number
9
Number of Choices for the Third Number
8
Number of Choices for the Fourth Number
7
Now we will use the Fundamental Counting Principle to find the desired number of possible combinations.
10 * 9 * 8 * 7 = 5040
Miranda has 5 40 different number combinations to determine the lock with the given restriction.
