Problem Solving

AAPL wrote:Three gnomes and three elves sit down in a row of six chairs. If no gnome will sit next to another gnome and no elf will sit next to another elf, in how many different ways can the elves and gnomes sit?

A. 18
B. 36
C. 48
D. 72
E. 96

The OA is D.

I solved this PS question as follows,

Case 1: G E G E G E - Total ways = 3!*3! = 36

Case 2: E G E G E G - Total ways = 3!*3! = 36

Total ways of arrangements = 36 + 36 = 72

Option D.

Has anyone another strategic approach to solve this PS question? Regards!

Your solution is perfect. It's also how I would have solved it.

Cheers,
Brent

Hi AAPL,

Your approach to this prompt is spot-on! With a small adjustment to your thinking, there's a way to perform all of the math in one calculation. We're given the specific rule that the seating will have to be either GEGEGE or EGEGEG.

The first chair can have EITHER a gnome or an elf, so there are 6 options:

6 _ _ _ _ _

Whether it's a gnome or an elf, the next chair must be someone from the other group (and there will be 3 options):

6 3 _ _ _ _

Then the remaining 4 chairs will have to go back-and-forth GEGE or EGEG. The third and fourth chairs will have 2 options each while the 5th and 6th chairs will have just one option for each. This gives us....

6 3 2 2 1 1

Thus, there are (6)(3)(2)(2)(1)(1) = 72 possibilities.

Final Answer: D

GMAT assassins aren't born, they're made,
Rich

AAPL wrote:Three gnomes and three elves sit down in a row of six chairs. If no gnome will sit next to another gnome and no elf will sit next to another elf, in how many different ways can the elves and gnomes sit?

A. 18
B. 36
C. 48
D. 72
E. 96

We can see that one seating arrangement of gnomes (G) and elves (E) can be:

GEGEGE

The first G and E each has 3 choices; the second G and E each has 2 choices and the last G and E each has 1 choice. Thus the number of ways to have the "GEGEGE" seating arrangement is:

3 x 3 x 2 x 2 x 1 x 1 = 36

However, the seating arrangement can also be EGEGEG, and there will also be 36 such arrangements. Thus, the total number of seating arrangements is 36 + 36 = 72.

Answer: D