a) How many arrangements using all of the letters of the word parallel are possible? b) How many

step 1 So in this question there are two parts. And the first part is asking us how many different arra

A) How many arrangements using all of the letters of the...

Key Concepts

Permutations with Repetition

This concept deals with arranging a multiset of objects where some items are identical. Instead of simply calculating the total number of arrangements with n!, it is necessary to divide by the factorial of the number of occurrences of each repeated object to correct for overcounting. In problems involving words or sets with repeating elements, this method ensures that each unique arrangement is only counted once.

Block Method (Grouping Items in Permutations)

The block method is used to count arrangements when a subset of items must appear together consecutively. By treating the grouped items as a single unit, one first calculates the number of ways to arrange this 'block' along with the other items and then multiplies by the internal arrangements of the items in the block if they are distinct. This approach simplifies the counting process under additional constraints.

Transcript

00:01 So in this question there are two parts.

00:03 And the first part is asking us how many different arrangements there are using all of the letters of the word parallel.

00:10 So first we can see that there are eight different letters in the word parallel.

00:14 So there will be eight factorial different ways to arrange all of them.

00:18 And we can do this by imagining that we have four or eight slots and then we have to pick a letter to put in each one.

00:24 So once we've used a letter we obviously cannot use it again.

00:27 So in the first slot we have eight possibilities, then seven, then the first slot we have eight possibilities, then seven, then six five four three two one so that's how we get eight factorial now we have to also account for repetition for example having two ls right next to each other one way does not affect if we switch these ls arounds because we would still have the same word and we can see that we have three different ls so that's three factorial ways that actually would be exactly the same since if we imagine three slots filled with the l, flipping these around does not change anything.

01:03 Second, we can also see that we have two a's here.

01:06 So for the same reason as the l's, we would have two factorial ways where the a's could be either together or apart and the order of the a's does not really matter.

01:15 So now we just have to evaluate this expression so we can simplify down the 8 factorial to 8 times 5, times 7, 10, 6, times 5 times 4 times 3 factorial, then 3 factorial times 2 factorial.

01:29 So 3 factorials can cancel out.

01:32 2 factorial is just 2 times 1.

01:34 So we can divide that into the 4.

01:36 So we get 8 times 7 times 6 times 5 times 2...

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