How many 3 letter word sequences can be formed using the letters A B C D if no letter is to be repeated?

This section covers permutations and combinations.

Arranging Objects

The number of ways of arranging n unlike objects in a line is n! (pronounced ‘n factorial’). n! = n × (n – 1) × (n – 2) ×…× 3 × 2 × 1

Example

How many different ways can the letters P, Q, R, S be arranged?

The answer is 4! = 24.

This is because there are four spaces to be filled: _, _, _, _

The first space can be filled by any one of the four letters. The second space can be filled by any of the remaining 3 letters. The third space can be filled by any of the 2 remaining letters and the final space must be filled by the one remaining letter. The total number of possible arrangements is therefore 4 × 3 × 2 × 1 = 4!

• The number of ways of arranging n objects, of which p of one type are alike, q of a second type are alike, r of a third type are alike, etc is:

n!        .
p! q! r! …

Example

In how many ways can the letters in the word: STATISTICS be arranged?

There are 3 S’s, 2 I’s and 3 T’s in this word, therefore, the number of ways of arranging the letters are:

10!=50 400
3! 2! 3!

Rings and Roundabouts

• The number of ways of arranging n unlike objects in a ring when clockwise and anticlockwise arrangements are different is (n – 1)!

When clockwise and anti-clockwise arrangements are the same, the number of ways is ½ (n – 1)!

Example

Ten people go to a party. How many different ways can they be seated?

Anti-clockwise and clockwise arrangements are the same. Therefore, the total number of ways is ½ (10-1)! = 181 440

Combinations

The number of ways of selecting r objects from n unlike objects is:

Example

There are 10 balls in a bag numbered from 1 to 10. Three balls are selected at random. How many different ways are there of selecting the three balls?

10C3 =10!=10 × 9 × 8= 120
             3! (10 – 3)!3 × 2 × 1

Permutations

A permutation is an ordered arrangement.

• The number of ordered arrangements of r objects taken from n unlike objects is:

nPr =       n!       .
          (n – r)!

Example

In the Match of the Day’s goal of the month competition, you had to pick the top 3 goals out of 10. Since the order is important, it is the permutation formula which we use.

10P3 =10!
            7!

= 720

There are therefore 720 different ways of picking the top three goals.

Probability

The above facts can be used to help solve problems in probability.

Example

In the National Lottery, 6 numbers are chosen from 49. You win if the 6 balls you pick match the six balls selected by the machine. What is the probability of winning the National Lottery?

The number of ways of choosing 6 numbers from 49 is 49C6 = 13 983 816 .

Therefore the probability of winning the lottery is 1/13983816 = 0.000 000 071 5 (3sf), which is about a 1 in 14 million chance.

Video transcript

- [Voiceover] So let's ask ourselves some interesting questions about alphabets in the English language. And in case you don't remember and are in the mood to count, there are 26 alphabets. So if you go, "A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T, U, V, W, X, Y, and Z," you'll get, you'll 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 we're not going to be thinking about phonetics or how hard it is to pronounce it. So, for example, the word, the word ZGT would be a legitimate word in this example. Or the word, the word, the word SKJ would be a legitimate word in this example. So how many possible three letter words are there in the English language? I encourage you to pause the video and try to think about it. Alright, I assume you've 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 ask this to you to be a distractor because we've seen a lot of examples. We're saying, "Oh, there's 26 possibilities "for the first one and 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 of the letters had to be different. So, for example, the word, the word HHH 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 time 26 and you can 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 to 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 think a lot of things like this, it's always best to reason through than try to figure out if some formula applies to it. So in this situation, well, if we went in 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 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 25 possibilities for what we put in the middle one if we say we're going to 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 I went 26, 25, 24 is to show you there's nothing special about always filling in the left most letter or the left most chair first. It's just about, well, let's just think in terms of let's just 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 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 what I have been doing. 26 possibilities for the left most 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 already used one letter and they have to be different. And then for the last bucket, you're going to 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, sorry, not two p. 20, my brain is malfunctioning. 26, we're figuring out how many permutations are there for putting 26 different things into three different spaces and this is 26, if we just blindly apply the formula, which I never suggest doing. It would be 26 factorial 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 so the whole point of this video, there's two points, is one, as soon as someone says, "How many different letters could you form" or something like that, you don't just 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 times 26. The other thing I want to point out, and 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 there's this formula "that I remember from weeks or years ago "in my life that had an N factorial and K factorial "and I had to memorize it, I have to look it up." Always much more useful to just reason it through.

How many 3

How many three letter words can be made using the letters A B C D E if no repetitions are allowed?

How many words can be formed using ABCD?

How many 3