How many vertically striped flags can be created if the flag

Question

How many vertically striped flags can be created if the flag can have up to 4 stripes and no color can be used more than once?

A) 24
B) 48
C) 60
D) 64
E) 256

Bunuel · Answer

Hi, and welcome to Gmat Club!

I guess the stem misses the part saying that we have only 4 colors to use. If so than:

P^1_4+P^2_4+P^3_4+P^4_4=4+12+24+24=64 :

P^1_4  - # of different flags with one stripe (eg: just green or yellow or red or black);

P^2_4  - # of different flags with 2 stripes;

P^3_4  - # of different flags with 3 stripes;

P^2_4  - # of different flags with 4 stripes.

We use P (permutation) as order of the stripes on the flag matters: yellow-black-red flag is different from red-black-yellow flag.

Answer: D.

Hope it's clear.

TallJTinChina · Answer

Do you mean  P^4_1+P^4_2+P^4_34+P^4_4=4+12+24+24=64  ?

Just read walker's post on Permutations from the Math Book and I am confused here.

I thought  P^1_4= 1! / (1-4)!  and so on.

And  P^4_1 = 4! / (4-1)! = 4

Bunuel · Answer

P^2_4  and  P^4_2  are the same (just different way of writing) and equal to  \frac{4!}{(4-2)!}  - the # of ways to choose 2 different objects out of 4, when order matters.

The expression you wrote,  P^1_4=\frac{1!}{(1-4)!}=\frac{1}{(-3)!} , does not make any sense for two reasons: 1.  P^n_k  (or which is the same  P^k_n ) gives the # of ways of choosing  n  different objects out of  k , when order matters, hence it's always an integer and 2. factorial of negative number is undefined.

Hope it helps.

TallJTinChina · Answer

So when we are solving we just put  \frac{{bigger!}}{{smaller!(bigger-smaller)!}}

Bunuel · Answer

For C yes:  C^n_k=\frac{{bigger!}}{{smaller!(bigger-smaller)!}} . For P:  P^n_k=\frac{{bigger!}}{(bigger-smaller)!} .

AndreG · Answer

Thanks a lot! If we can only use 4 colours this makes sense, must be missing info in the quesiton stem as you said! Kudos to you! :-)

IIMC · Answer

How could this question can be answered when it' not know that how many colors are there ?

Bunuel · Answer

Please check here: https://gmatclub.com/forum/how-many-ver ... ml#p749411

ScottTargetTestPrep · Answer

Since the question states the flag can have up to 4 stripes, the answer will equal #(flags with 1 stripe) + #(flags with 2 stripes) + #(flags with 3 stripes) + #(flags with 4 stripes).

#(flags with 1 stripe):

Since we have 4 colors to choose from, there are simply 4 flags with one vertical stripe.

#(flags with 2 stripes):

We note that this is a permutation problem since the order matters (i.e., a blue-red striped flag is not the same thing as red-blue striped flag).

Since we have 4 available colors and we need 2 colors, the number of flags with 2 stripes is 4P2 = 4!/2! = 4 x 3 = 12.

#(flags with 3 stripes):

Since we have 4 available colors and we need 3 colors, the number of flags with 3 stripes is 4P3 = 4!/(4-3)! = 4 x 3 x 2 = 24

#(flags with 4 stripes):

Since we have 4 available colors and we need 4 colors, the number of flags with 4 stripes is 4P4 = 4!/(4-4)! = 4 x 3 x 2 x 1 = 24

In total, there are 4 + 12 + 24 + 24 = 64 available choices.

Answer: D

prakhar992 · Answer

Hi Bunnel,

I hope that you are doing well in this time of pandemic. 
I solved this question but got confused.
Suppose, we want to solve this question by logic and not by formulae.

There are 4 colors and 4 stripes, First, we will pick one color. 4 ways, then we will pick one stripes 4 ways, then we will pick one color.3 ways, then we will pick one stripes 3 ways, then we will pick one color. 2 ways, then we will pick one stripes 2 ways, then we will pick one color. 1 ways, then we will pick one stripes 1 ways.

Total number of ways - 4*4*3*3*2*2*1*1 = 24*24 =576. 
I know that i am missing something. Can you please tell me what i am missing.

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How many vertically striped flags can be created if the flag

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