# Question Video: Applications of the Counting Principle (Product Rule) Mathematics

A code breaker is trying to find the value of an eight digit number. The figure below shows the digits that he has already discovered. He has narrowed down his options for the digit represented by the letter 𝑐 to the following set of numbers {5, 6, 4}. Given that he currently knows nothing about the other digits, how many possible numbers does he have left to try?

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### Video Transcript

A code breaker is trying to find the value of an eight-digit number. The figure below shows the digits that he’s already discovered. He’s narrowed down his options for the digit represented by the letter 𝑐 to the following set of numbers: five, six, four. Given that he currently knows nothing about the other digits, how many possible numbers does he have left to try?

In this question, we are looking at the number of possible outcomes for four events, the four remaining digits of the code. In order to work out how many possible outcomes there are, we need to recall the fundamental counting principle. This tells us that the number of possible outcomes in a sample space can be found by multiplying the number of possible outcomes from each event.

So let’s look at the number of possible outcomes for each event. In this question, that’s each remaining digit in the code. We’re told that the digit represented by the letter 𝑐 has been narrowed down to either five, six, or four which means there are three possible outcomes for this event. We know nothing about the other digits which means they could be any of the numbers from zero to nine. And therefore, there are 10 possibilities for each of the other digits. The fundamental counting principle, remember, tells us that we can multiply the number of possible outcomes from each event. Therefore, the total number of possible outcomes is found by multiplying three by 10 and then by 10 and then by 10 again.

Therefore, the total number of possibilities that the code breaker needs to try is 3000.