Flag

[shashank.ism]

A new flag is to be designed with six vertical stripes using some or all of the colours yellow, green, blue and red. Then, the number of ways this can be done such that no two adjacent stripes have the same colour is

a) 12x81
b) 16 x 192
c) 20x 125
d)24x216
e) None of these

[harsh.champ]

A new flag is to be designed with six vertical stripes using some or all of the colours yellow, green, blue and red. Then, the number of ways this can be done such that no two adjacent stripes have the same colour is

a) 12x81
b) 16 x 192
c) 20x 125
d)24x216
e) None of these

Now,4 colors are there .
1st stripe with suppose color A.[this can be selected in 4 ways]
2nd stripe can be B,C,D.
Correspondingly,3rd stripe will be A,B,C.(suppose D in the 2nd stripe)
Correspondingly,4th stripe will be D.,B,C.(suppose A in the 2nd stripe)
Like this for the 5th and 6th stripe.
Ans. will be [spoiler]4 x 3^5 = 972 A.[/spoiler]

[Skbk07]

In this question why we are not focus ed on the words "some" and "or" means all the time all colours need not use to make the new flag . it may be randomly selected two or three or four or five or all of six.

[DavidG@VeritasPrep]

In this question why we are not focus ed on the words "some" and "or" means all the time all colours need not use to make the new flag . it may be randomly selected two or three or four or five or all of six.

You're correct, but the answer does consider these possibilities.

The first stripe can be any of the four colors. Stripe 1: 4 options

Let's say, for illustrative purposes that the first stripe is yellow. Stripe 2 can be green, blue, or red. Stripe 2: 3 options

Say stripe 2 is green. Now stripe 3 can be yellow, red, or blue: 3 options.

To summarize

Stripe 1: 4 options
Stripe 2: 3 options (we exclude whatever we picked for stripe 1)
Stripe 3: 3 options (we exclude whatever we picked for stripe 2)

The same logic will hold for all of the remaining stripes. If the next stripe must be a different color than the preceding one, there will be 3 options to choose from. So the answer is 4 * 3^5, which can be rewritten as 4 * 3 * 3^4 or 12 * 81.

Notice that the above framework contains the possibility that only two colors are used. (For example Yellow-Green-Yellow-Green-Yellow_Green is among those 4 *3^5 possibilities. And Green-Blue-Red-Green-Blue-Red is also among them. There are also could have been 4 colors selected. The only restriction is that each stripe must be different from the adjacent stripes.)

[[email protected]]

Hi Skbk07,

The prompt as us for the TOTAL number of possible ways to create a flag given that there will be 6 'stripes' and up to 4 possible colors, with the added condition that no two adjacent stripes have the same color. While you COULD break the math down into the various possibilities (just 2 colors, just 3 colors or all 4 colors), that would be an inefficient way to approach the question (as it would require lots of 'excess math'). By its design, David's approach includes every possible outcome - and it's the most efficient way to get to the correct answer.

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

[Matt@VeritasPrep]

In this question why we are not focus ed on the words "some" and "or" means all the time all colours need not use to make the new flag . it may be randomly selected two or three or four or five or all of six.

"some or all" is actually redundant here: all we need to do is say "some of the colors". Since we can't have all six stripes differ (we only have four colors to choose from), we know that we're going to be forced to repeat at some point. Beyond that, the problem would only need to explicitly stipulate that we use ALL four colors; if that isn't stipulated, we assume that, say, RYRYRY is a fine flag.