Tom, Bill, Robert, Roger, and Terry are standing in a row for a group

different orders possible for the five men, if Tom refuses to stand next to Roger = total number of orders - orders with Tom and Roger are together
= 5! - 4!*2!

(consider Tom and Roger as a single person, so we have to find the number of combinations for 4 which is 4!
and Tom and Roger can be together in 2! ways - Tom and Roger or Roger and Tom)

=120 - 48
=72

Hence Option C is correct
Hit Kudos if you liked it

Answer : [C]

Total Ways : 5!
Ways where they stand together = 4! * 2 [4! considering both as a single unit and multiplied by 2 for internal switching of arrangement]

Ways where they dont stand together = 5! - 4!*2 = 72

Here's another approach:

Take the task of arranging the 5 men and break it into stages.

Stage 1: Arrange Bill, Robert, and Terry in a row
There are 3 people, so we can arrange them in 3! ways.

Now that we've arranged 3 men, we'll place a potential standing space on each side of these 3 men.
For example: ___ Terry ___ Robert ___ Bill ___
Notice that, when we place the 2 remaining men (Tom and Roger), in the 4 available spaces, we will be guaranteed that they are not next to each other.

Stage 2: Select a place for Tom to stand
There are 4 spaces available,, so we can complete this stage in 4 ways.

Stage 3: Select a place for Roger to stand
There are 3 remaining spaces, so we can complete this stage in 3 ways.

By the Fundamental Counting Principle (FCP), we can complete all 3 stages (and thus arrange all 5 men) in (3!)(4)(3) ways (= 72 ways)

Answer: C

Note: the FCP can be used to solve the MAJORITY of counting questions on the GMAT. For more information about the FCP, watch this video from our course:



You can also watch a demonstration of the FCP in action here:

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Dear GMATPrepNow, Could you please help to elaborate the highlighted portion? I have no idea how do you able to arrange in 4 ways & 3 ways.
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Sure thing.
Once we've places spaces on each side of the 3 people who have already been place (e.g., ___ Terry ___ Robert ___ Bill ___ ), we have 4 distinct locations to place the 2 remaining men.
They are: __1__ Terry __2__ Robert __3__ Bill __4__

So...
Stage 2: Select a place for Tom to stand
There are 4 spaces available.
So, we can place Tom in space #1 (above), OR space #2 (above), , OR space #3 , OR space #4
Since we have 4 options, we can complete this stage in 4 ways.

Stage 3: Select a place for Roger to stand
In stage 2, we placed Tom in one of the 4 available spaces.
This means there are only 3 remaining spaces in which to place Roger.
Since we have 3 options remaining, we can complete this stage in 3 ways.

Does that help?

Cheers,
Brent
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We are given that 5 people must stand in a row for a photo and need to determine in how many different orders they can stand if Tom refuses to stand next to Roger.

We can create the following equation:

total # of arrangements = # ways Tom is next to Roger + # ways Tom is NOT next to Roger.

Thus:

total # of arrangements - # ways Tom is next to Roger = # ways Tom is NOT next to Roger.

Let’s determine the total number of arrangements and the number of ways Tom is next to Roger.

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

The total number of arrangements with Tom next to Roger can be calculated as follows:

[Tom-Roger] - [Bill] - [Terry] - [Robert]

Since Tom must be with Roger, notice there are 4! ways to arrange the entire group. However, we must remember that Tom and Roger can be arranged in 2! ways since it could be [Tom-Roger] or [Roger-Tom].

Thus, the number of ways Tom is next to Roger is 4! x 2! = 24 x 2 = 48.

Finally, the number of ways Tom is NOT next to Roger is 120 - 48 = 72.

Answer: C
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Tom, Bill, Robert, Roger, and Terry are standing in a row for a group photo. In how many different orders can the five men stand if Tom refuses to stand next to Roger?

A. 48
B. 64
C. 72
D. 96
E. 120
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# total number of different orders - # number of different orders with Tom and Roger standing together = # number of different orders with Tom and Roger not standing together.

5! - 4! x 2 (Tom and Roger can change their positions, hence 2 cases for that) = 120 - 48 = 72

The answer is 72, so C.

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