Use the fundamental counting principle to determine the total number of outcomes of choosing a number from two to 37 and a book from 14 books.
The fundamental counting principle, sometimes called the product rule for counting, is a technique that will allow us to count assortments of items. It tells us that if there are 𝑚 outcomes for one event and 𝑛 outcomes for another, then the total number of outcomes for both events combined is 𝑚𝑛. In other words, we simply multiply the number of outcomes for each event together. So, what are our two events?
Well, we see that event number one is choosing a number from two to 37. So, how many outcomes does this have? Well, since we’re choosing a number from two to 37 and we’re including both two and 37 in this count, there are 36 possible outcomes. Now, if you’re struggling to picture why it’s 36 and not 35, essentially, we’re counting the first 37 numbers, but not including the number one. So that’s 37 minus one, which is 36 outcomes.
Then, event number two is choosing one book from a possible 14. And so, the number of outcomes we have for the second event is 14. The fundamental counting principle or the product rule for counting tells us that the total number of outcomes for these events combined is the product of their respective outcomes. So, that’s 36 times 14. And of course, we could use a calculator to do this. But let’s recall how we might use the column method. We’re going to multiply each of the digits in the number 36 by the number four.
Six times four is 24, and we carry the two. Then, three times four is 12. 12 add two is 14. So we see that 36 times four is 144. We now repeat this process, but we multiply each of the digits in the number 36 by one. But of course, we’re really multiplying by 10. So, we add a zero in this first column here. Six times one is six, and three times one is three. So, 36 times 10 is 360. To find the product then of 36 and 14, we add these two three-digit numbers together. Four plus zero is four. Four plus six is 10. So, we carry the one. And then, one plus three plus that one is five. And so, we find 36 times 14 is equal to 504.
And so, we see the total number of outcomes of choosing a number from two to 37 and a book from 14 books is 504.