In how many ways can Ann, Bob, Chuck, Don and Ed be seated

Yes, your logic is absolutely fine. If the number of people is manageable, we can directly find the number of ways in which the two of them should not be seated together.
You arranged C, D and E in 3! ways.

The 4 dots show 4 positions for Ann.When Ann occupies one of these positions, 3 positions are left over for Bob. So Ann and Bob can sit in 4*3 = 12 ways.
Total 3! * 12 = 72 ways.

In these types of questions, always try to subtract the not allowed arrangements from the total arrangements. This makes our lives much easier.
Here we will assume that Ann and Bob always sit together i.e. they are a single entity
Total arrangements: 5!

Arrangements in which Ann and Bob sit together = 4!*2!
We get this because, total arrangements =4! and Ann and Bob can be arranged in 2! ways

Subtracting this from the total: 5! - 4!*2! = 72. Option D

We can use the following formula:

Total number of ways to arrange the 5 people = (number of arrangements when Bob sits next to Ann) + (number of arrangements when Bob does not sit next to Ann)

Let’s determine the number of arrangements when Bob sits next to Ann.

We can denote Ann, Bob, Chuck, Don, and Ed as A, B, C, D, and E, respectively.

If Ann and Bob must sit together, we can consider them as one person [AB]. For example, one seating arrangement could be [AB][C][D][E]. Thus, the number of ways to arrange four people in a row is 4! = 24.

However, we must also account for the ways we can arrange Ann and Bob, that is, either [AB] or [BA]. Thus, there are 2! = 2 ways to arrange Ann and Bob.

Therefore, the total number of seating arrangements is 24 x 2 = 48 if Ann and Bob DO sit next to each other.

Since there are 5 people being arranged, the total number of possible arrangements is 5! = 120.

Thus, the number of arrangements when Bob does NOT sit next to Ann is 120 - 48 = 72.

Answer: D

This is a classic FCP with restrictions problem. It's faster to use the "complement" to solve these problems than to count the arrangements directly. If we can count the number of total arrangements, then subtract the number of cases we don't want, we're left with the number of cases we do want.

[# of arrangements with Ann & Bob not sitting together]

= [# without restrictions] - [# with Ann & Bob sitting together]

When arranging 5 people where order matters, the # of arrangements is 5! = 120.

To calculate the # of arrangements where Ann & Bob are seated together, we "glue" them together as a single unit: [Ann, Bob]. In that case we're arranging 4 units where order matters, that's 4! = 24.

But wait! Ann, Bob, Chuck, Don, Ed is different from Bob, Ann, Chuck, Don, Ed even though Ann and Bob are still seated together before the rest. So, there's an additional 24 arrangements where we have them glued together as [Bob, Ann].

The # of arrangements with Ann & Bob not sitting together is 120 - 24 - 24 = 72.

For a review, on this "complement" concept check out the FCP with Restrictions lesson video.

Total ways to sit = 5! =120
Ann and Bob sitting together = 4!* 2(Ann in left, Bob right or Bob left, Ann in right) = 48
Required ways = 120 -48 = 72

D is the answer.

I solved this question with the below logic:
I assumed 5 blanks that will be filled by all the 5 and place A & B apart as shown below.
_ A_B_

Now the blank spaces will be filled by C,D,E in 3! ways

A&B can be rearranged in 2! ways

And assuming A_B as X we have _X_. We can rearrange these in 3! ways

So the total ways of A and B are not seated together are 3! X 2! X 3! = 72

Can some one please guide whether this approach is correct?

Thank You,
Regards,
Shashank Singh

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