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Numerade Educator

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Problem 26 Hard Difficulty

How many monograms consisting of three initials are possible?

Answer

17576

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Video Transcript

this example will show us how to find the total number of different monograms that could be created. A monogram, if you were not sure, is just your first middle and last initial. So it's a three letter sequence. Those letters could possibly be anything. For example, my initials would be mjd Michael Jandola. They could use the same letter more than once. So, for example, maybe your name is Jeremy Jason Jones. So it doesn't have to be a different letter. Each time, names could be repeated the same initial. And I'm gonna illustrate how this works before I just show you the math really fast. I'm gonna do this by showing you a sample space in the form of a probability tree, for example. Our first initial could be a could be be could be c could be d It could be anything all the way up, Dizzy. So our first initial there's 26 possibilities. No, pretend our first initial was a and we want to go and say, Hey, how many possibilities other for a middling? While the middle name could still be within a or a B or with any initial all the way up dizzy again, so each of those initial 26 branches had 26 branches coming off of them. So just to get a first in the middle initial would be 26 times 26 last. We've got to imagine, like, How would we get that third initial? Well, let's pretend that the initials were a and then B, and now we've got to say, how many options are there? How many possibilities for that last initial? While you would still have every letter of the alphabet as a possibility. So 26 branches to start with. Those are the ones you saw right there, 26 branches to start with off of those 1st 26 branches, each of those would have had 26 branches and then off of each of those middle initials you have 26 more for that last initial. It would be really, really tedious to draw the whole thing. Matter of fact, if you do 26 times 26 times 26 or 26 to the third power, you would get a result of 17,576

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