If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

Main content
Current time:0:00Total duration:5:39

Video transcript

so let's ask ourselves some interesting questions about alphabets in the English language and in case you don't remember and the mood to count there are 26 alphabets so if you go a b c d e f g h ng canna pqrstuvwxyz you get you get 26 26 alphabets now let's ask some interesting questions so given that there are 26 alphabets in the English language how many possible three-letter words are there and and we're not gonna be thinking about phonetics for how hard it is to pronounce it so for example the word the herd ziget would be a legitimate word this example or the word the word the word sched would be a legitimate work if it's example so how many possible three-letter words are there in the English language encourage you to pause the video and try to think about it alright I assume you had a go at it so let's just think about it for three letter words there's three spaces so how many possibilities are there for the first one well there's 26 possible letters for the first one anything from A to Z would be completely fine now how many possibilities for the second one and I intentionally asked this to you to be a distractor because you might be you know we've seen a lot of examples we're saying oh there's 26 possibilities for the first one and then maybe there's 25 for the second one and then 24 for the third but that's not the case right over here because we can repeat letters I didn't say that all the letters had to be different so for example the word the word would also be a legitimate word in our example right over here so we have 26 possibilities for the second letter and we have 26 possibilities for the third letter so we're going to have and I don't know what this is 26 to the third power possibilities or 26 times 26 times 26 and you could figure out what that is that is how many possible three-letter words we can have for the English language if we didn't care about how pronounceable they are if they meant anything and if we repeated letters now let's ask a different question what if we said how many possible three-letter words are there if we want all different letters so we want all different letters so these all have to be different letters different different letters and once again pause the video and see if you can think it through alright so this is where permutations start to be useful although I you know I think a lot of things like this it's always best to reason through than to try to figure out whether some formula applies to it so in this situation well if we went an order we could have 26 different letters for the first one 26 different possibilities for the first one you know I'm always starting with that one but I there's nothing special about the one on the left we could say that the one on the right there's 26 possibilities well for each of those possibilities for each of those 26 possibilities there might be twenty-five possibilities for what we put in the middle one if we say we're gonna figure out the middle one next and then for each of these 25 times 26 possibilities for where we figured out two of the letters there's 24 possibilities because we've already used two letters for the last bucket that we haven't filled and the only reason why I went where I went 26 25 24 is to show you there's nothing special about always filling in the what the leftmost letter or the leftmost chair first it's just about well let's just think in terms of let's fill out one of the buckets first hey we have the most possibilities for that once we use something up then for each of those possibilities we'll have one left one less for the the the next the next bucket and so I could do 24 times 25 times 26 but just so I don't fully confuse you I'll go back to the what I have been doing 26 possibilities for the leftmost one for each of those you would have 25 possibilities for the next one that you're going to try to figure out because you've already used one letter and they have to be different and then for the last bucket you're gonna have 24 possibilities so this is going to be 26 times 25 times 24 whatever that happens to be and if we wanted to write it in the notation of permutations we would say that this is equal to we're taking 26 things signed up to P 20 my brain is malfunctioning 26 we're taking on how many permutations are there for putting 26 different things into three different space and this is 26 if we just blindly apply the formula which I never suggest doing it be 26 electorial over 26 minus three factorial which would be 26 factorial over 23 factorial which is going to be exactly this right over here because the 23 times 22 times 21 all the way down to one is going to cancel with the 23 factorial and and so the whole point of this video there's two points is one you know what as soon as someone says hello how many different letters could you form or something like that you just don't blindly do permutations or combinations you think about well what is being asked in the question here I really just have to take 26 times 26 26 the other thing I want to point out I know I keep pointing it out and it's probably getting tiring to you is even when permutations are applicable in my brain at least it's always more valuable to just try to reason through the problem as opposed to just saying oh my there's this formula that I remember from weeks or years ago in my life that had n factorial and K factorial and have to memorize and have to look it up always much more useful to just reason it through