Find the number of permutations of the letters in the word Honolulu if vowels and consonants must be kept together

Question

I honestly don't understand questions like this. I get the main concept of permutations, but then when I see a question like this, my brain just freezes. Are there any steps I can go through for any permutation problem that will always help me figure out the right answer?

Answer 1

Some permutation problems can be looked at as straight-forward permutation problems with a restriction. For example, maybe I want to rearrange the letters in the word CANDY so that the Y is the third letter. That's a lot like finding the number of ways you can rearrange CANDY (with no restrictions). In the latter problem (with no restrictions), imagine you've got 5 slots to put the letters C, A, N, D, and Y into:

 

_  _  _  _  _

 

You'll choose letters one by one for each slot, starting with the first. That means you have 5 possible letters to choose for the first slot, 4 letters remaining to choose for the second slot (since you've already chosen one), 3 letters for the third slot, 2 for the fourth slot, and 1 for the fifth slot:

 

_  _  _  _  _

5 4 3 2 1

 

Multiply those numbers to get the total number of ways to rearrange the letters in CANDY (with no restrictions): 5×4×3×2×1 = 120 ways

 

Now, let's get back to the problem with the restriction. I'd like to rearrange CANDY, but the Y must be in third slot. Start with the restriction. There's only one letter to choose for the third slot (Y):

 

_  _  _  _  _

       1

 

Then, I've got 4 letters remaining for the first slot, 3 for the second, 2 for the fourth, and 1 for the fifth:

 

_  _  _  _  _

4 3 1 2 1

 

Multiply those numbers to find the number of ways to rearrange CANDY with Y in the third slot: 4×3×1×2×1 = 24 ways

 

 

 

Some permutation problems ask you to take into account redundant answers. For example, if I wanted to rearrange the letters in the word APPLE, I could think through it like before:

 

_  _  _  _  _

5 4 3 2 1

 

I'd come up with 5×4×3×2×1 = 120 ways

 

But that's not quite right, because there are two P's. The word A P₁ P₂ L E is the same (for this problem) as the word A P₂ P₁ L E. So, we'll need to figure out how many ways there are to rearrange just the two P's and divide by that:

 

_ _

2 1

 

There are 2×1 = 2 ways to rearrange the P's. So, the number of ways to rearrange APPLE is actually 120/2 = 60 ways

 

 

 

Some permutation problems can be broken down into smaller problems. Yours is like that. So, to solve your problem, we'll break it down into pieces and use some of the methods above. I'll assume you want to make sure all the vowels end up together and all the consonants end up together. (HNLLOOUU would be okay, but HONOLULU would not.)

 

 

1. WAYS TO REARRANGE THE CONSONANTS

 

There are four consonants: H, N, L, and L. We can rearrange them like this:

 

_  _  _  _

4 3 2 1

 

4×3×2×1 = 24

 

But we've got a redundancy: there are two L's. As above, there are 2 ways you can arrange the two L's. So, there are 24/2 = 12 ways to rearrange the consonants.

 

 

2. WAYS TO REARRANGE THE VOWELS

 

There are four vowels: O, O, U, and U. We can rearrange them like the consonants:

 

4×3×2×1 = 24

 

But we've got two redundancies: there are two O's and two U's. So, divide by 2 for the O's to get 24/2 = 12 ways and divide by 2 for the U's to get 12/2 = 6 ways to rearrange the vowels.

 

(That's very few ways, so you can actually check your answer. There are indeed only 6 ways to arrange those letters: OOUU, OUOU, OUUO, UOOU, UOUO, and UUOO.)

 

 

3. PUT THAT TOGETHER

 

So far, there are 12 ways to arrange the consonants. For every way to arrange the consonants, there are 6 ways to arrange the vowels. That's a total of 12×6 = 72 ways.

 

There's just one more thing to take into account: I can put either the group of vowels (V) or the group of consonants (C) first. This is basically another mini-permutation problem. I've got two things (V and C) and I want to know how many ways there are to arrange them:

 

_ _

2 1

 

As before, there are 2×1 = 2 ways to arrange the two things. So, for every one of the 72 ways to arrange the vowels and the consonants within their groups, there are 2 ways to arrange the groups. That's a total of 72×2 = 144 ways to rearrange the letters in HONOLULU, grouping the vowels and the consonants.

 

 

 

I know that's a lot to take in... Hopefully this helps you figure out how to approach different permutation problems in the future! Leave a comment if anything isn't clear.

 

 

 

Just as a quick practice problem, try to figure out how many ways there are to arrange the letters in the word BANANA. (You'll have to arrange the letters and take into account the redundant A's and N's, and you should come up with 60 ways.)

 

What if I want the B to be the third letter? (You'll have to start with the restriction, arrange the other letters, and take into account the redundant A's and N's. You should come up with 10 ways.)

 

What if I want the N's to be next to each other, and I don't care where the other letters are? (You'll want to figure out how many ways there are to arrange the other four letters, taking into account the redundant A's. Then, multiply that by how many places there are to put the N's if they're next to each other. You can put the N's in slots 1 and 2, 2 and 3, 3 and 4, 4 and 5, or 5 and 6. You should come up with 20 ways.)

 

Finally, what if I want the N's to be grouped and the A's to be grouped? (Try to figure out why there are only 6 ways.)