Possible Arrangements (with condition) - Soccer Team Flags

[carlos.lara.7]

A child received a gift of six different soccer team flags, including Liverpool and Arsenal. If he only has space in his bedroom to display four flags in a row, how many arrangements are possible if he cannot display the Liverpool and Arsenal flags at the same time?

a) 162
b) 216
c) 272
d) 360
e) 414

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Hi carlos.lara.7,

There are a couple of different ways to approach this question, but they all require a certain amount of 'math work.'

We're given 6 flags and asked for the number of different arrangements of 4 flags. IF there were no other restrictions, then the total would be...

(6)(5)(4)(3) = 360

However, the flags for Liverpool and Arsenal CANNOT be displayed at the same time, so we have to remove some of those 360 possibilities. The easiest way to do that is to total up the number of possible ways that the Liverpool and Arsenal flags COULD be displayed together; then we'll just subject that number from 360.

To start, let's say we had those 2 flags and 2 of the remaining 4 flags. With those 4 flags, there would be...

(4)(3)(2)(1) = 24 arrangements

Since there are 4 'other' flags to choose from, there are 4c2 = (4)(3)(2)(1)/(2)(1)(2)(1) = 6 different pairs of 2 other flags that could be grouped with Arsenal and Liverpool.

(6)(24) = 144 possible arrangements that have Arsenal and Liverpool (which we have to remove from the 360):

360 - 144 = 216

Final Answer: B

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

[DavidG@VeritasPrep]

A child received a gift of six different soccer team flags, including Liverpool and Arsenal. If he only has space in his bedroom to display four flags in a row, how many arrangements are possible if he cannot display the Liverpool and Arsenal flags at the same time?

a) 162
b) 216
c) 272
d) 360
e) 414

I'd probably solve this one the way Rich did, but one alternative is to add up all the desired cases.

Case1: Include Liverpool but not Arsenal. First we need to choose three other flags (from the remaining four) to go along with the Liverpool flag. There are 4 ways we can do this. (If we're choosing 3, we're leaving off one, and there are four different ways we can leave off one flag.) Once we have our 4 flags, there are 4! ways to arrange them. So Case 1: 4*4! = 96

Case 2: Include Arsenal but not Liverpool. Math is identical to Case 1. So another 96 options.

Case 3: Don't choose Arsenal or Liverpool. So we know we're choosing the other four flags. There are 4! = 24 ways to arrange them

Add 'em up: 96 + 96 + 24 = 216. Answer is B

[Matt@VeritasPrep]

We could also do

Total - Invalid

Total here = 6 * 5 * 4 * 3 => 360 ways of arranging four flags.

Invalid = 2 * 1 * 4 * 3 * (4 choose 2) => 144 invalid arrangements. (That's 2*1 for Liverpool and Arsenal in some order, then 4*3 for two of the four other flags, then (4 choose 2) because we need to choose two spaces of the four in which to place the Liverpool and Arsenal flags.)

So we've got 360 - 144 = 216 possibilities.

[Scott@TargetTestPrep]

A child received a gift of six different soccer team flags, including Liverpool and Arsenal. If he only has space in his bedroom to display four flags in a row, how many arrangements are possible if he cannot display the Liverpool and Arsenal flags at the same time?

a) 162
b) 216
c) 272
d) 360
e) 414

We are given that a child received 6 different soccer team flags, including Liverpool and Arsenal. We need to determine the number of arrangements that are possible when the flags are displayed 4 at a time and the Arsenal and Liverpool flags are not displayed at the same time.

To start we can create the following equation:

Total number of ways to display the flags = (number of ways to display the flags when both the Arsenal and Liverpool flags are displayed together) + (number of ways to display the flags when the Arsenal and Liverpool flags are not displayed together).

Let's first determine the total number of ways to display the 4 flags from a choice of 6 flags.

This is a permutation problem because the order in which the flags are displayed is important.

Number of ways to display the 4 flags from 6 flags = 6P4 = 6 x 5 x 4 x 3 = 360

Next we can determine the number of ways to display the flags when both Arsenal and Liverpool are displayed. Since we know that the Arsenal and Liverpool flags are definitely selected, that leaves us with 4 flags for the 2 remaining spots, so there are 4C2 ways to select the two remaining flags, which equals (4 x 3)/2! = 6 ways. Finally, there are 4! ways to arrange those 4 flags, which equals 24 ways. Thus, there are 24 x 6 = 144 ways to select and arrange the flags in which both Arsenal and Liverpool are displayed.

Finally, there are 360 - 144 = 216 ways to select and arrange the 4 flags when Arsenal and Liverpool cannot be displayed at the same time.

Answer: B