The Fundamental Counting Principle

in this lesson we're going to talk about the fundamental counting principle this principle can help us to determine the total number of possible outcomes that can occur in situation and the way we get this number is by multiplying the number of outcomes that occur for each event within that situation so let's look at this problem or situation that we're dealing with how many different outfits can Mike have with two pants three shirts and two pairs of boots so this situation or problem has three events the first event has to do with his choice of pants that he's going to wear the second event has to do with which shirt he's going to put on and the third event has to do with which boot or pairs of boots that he's gonna wear so in the first event he has two choices he has two pants to wear for the second event he has three shirts and for the third event he has two boots so if we multiply these numbers 2 times 2 times 3 we're going to get 12 possible outfits that he can wear and that's the answer now to illustrate how this works we could use tree diagram to get the same answer so for his first choice he has two pants that he can wear either white or black now for her second choice he has three shirts that he can choose from I'm going to color code this so he could choose red shirt he could choose green shirt or he can choose an orange shirt and the same is true if he were to wear pair if he were to wear black pants he can still pair that up with red shirt green shirt or an orange shirt now for his next joints we'll be what color boots he's going to wear and there's only two options here purple or yellow now if we follow the tree diagram we can get all the different possible outcomes he could wear white shirt mean white pants red shirt and purple boots so that's going to be or he could wear white shirt mean keep messing that up white pants red shirt or yellow boots which would be weird looking outfit but for the sake of this problem we'll work with it the next option could be wgp or next is and w-o-y now you can also choose black pants red shirt and purple boots or we could have b-r-y next we could have bgy mean bgp let's not skip that one and then bgy and then we can have so notice that what we have is total of 12 possible outcomes those are the different outfits that he can wear and that's how the fundamental counts and principle works all we do is simply multiply the number of outcomes that can occur for each event or each choice that he's going to make so for the First Choice he has there's two possible outcomes for the second choice there's three possible outcomes and for the third choice two so two times three times two gives us the 12 possible outcomes that we see for the entire situation now let's move on to the second problem how many different telephone numbers are possible in the U.S if the three-digit area code followed by the seven digit local telephone number cannot begin with 0 or 1. so here's an example of telephone number let's say three five seven four eight six one two seven nine so that's typical us telephone number so this is the three digit area code and what we have here is the seven digit local telephone number so we can't have zero or one in those two spots knowing that how can we determine how many different telephone numbers are possible in the US so let's start with the first digit so we have the numbers 0 1 2 3 4 5 6 7 8 9. that's ten possible numbers but we can't have zero or one so we have eight possible choices four the first spot now for the second spot we can have any number from zero to nine so that's ten possibilities and the same is true for the third spot now for this spot we can have zero or one so that's going to be eight possibilities everything else we can have all numbers from zero to nine so that's going to be 10 possibilities so now we need to do the math we're going to multiply 8 by 10 by 10 by 8 by 10 by 10 and by another four tenths so we're multiplying two-eighths that's 8 squared and we're multiplying that by eight tenths so that's 10 to the eighth power 8 squared is 64. and this is going to be times 10 to the 8. now let's put this in scientific notation 64 is 6.4 times 10. 10 is the same as 10 to the one so when we multiply these two we can add one and eight which will give us 9. so our answer is 6.4 times 10 to the nine ten to the 9 is billion so our final answer is 6.4 billion which you can write it like this if you want so that's how many different telephone numbers that are possible in the US with the restrictions that we can have zero or one four the first number for the area code and the first number for the local telephone number for the sake of practice let's work on one more question multiple choice quiz has four questions each with five answer choices in how many ways can you answer the questions so let's say this is the first test question this one's going to be the second this is the third that's the fourth so this situation has four events in the first event or the first test question you have five answer choices likely for the next question you also have five answer choices and the same is true for the third and the fourth question so you have five possible outcomes for each event and you have four events so this is going to be five to the fourth power or five times five times five times five times five is twenty-five and 25 times 25 is 625. so there's 625 ways in which you could answer the four test questions so that's it for this video hopefully gave you good introduction into the fundamental counter principle and how you could use it to answer questions like these