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John has to paint a wall with seven horizontal stripes. He only has en

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Bunuel
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John has to paint a wall with seven horizontal stripes. He only has en [#permalink] New post   15 Jan 2021, 11:36
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John has to paint a wall with seven horizontal stripes. He only has enough paint for four red stripes, four blue stripes, and four yellow stripes. If he can use at most two colors, how many different ways can he paint the wall?

(A) 420
(B) 210
(C) 84
(D) 35
(E) 7

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Re: John has to paint a wall with seven horizontal stripes. He only has en [#permalink] New post   15 Jan 2021, 14:46
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1. Number of ways to select 2 colors out of 3 = 3C2=3

2. Number of ways to select 1 color for 4 stripes and other for 3 stripes= 2

3. Total possible arrangements of 7 stripes\(= \frac{7!}{4!*3! }= 35\)

Number of ways in which he can paint the wall = 3*2*35= 210

Bunuel wrote:
John has to paint a wall with seven horizontal stripes. He only has enough paint for four red stripes, four blue stripes, and four yellow stripes. If he can use at most two colors, how many different ways can he paint the wall?

(A) 420
(B) 210
(C) 84
(D) 35
(E) 7

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petroshatz
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Re: John has to paint a wall with seven horizontal stripes. He only has en [#permalink] New post   15 Jan 2021, 15:36
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Starting we need to pick the colors that we will use.
so 3C2.

Since all colors can color at most 4 stripes, we know that the configuration of the two colors picked, say A and B, can only be 4 stripes color A and 3 stripes color B, or vice versa. It suffices to just pick the 4 stripes color A will paint.
7C4 but also need to do with the same with color B painting 4 stripes. So 7C4 X 2

so all together: 3C2 X 7C4 X 2 = 210
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Re: John has to paint a wall with seven horizontal stripes. He only has en [#permalink] New post   22 Feb 2021, 11:19
Bunuel wrote:
John has to paint a wall with seven horizontal stripes. He only has enough paint for four red stripes, four blue stripes, and four yellow stripes. If he can use at most two colors, how many different ways can he paint the wall?

(A) 420
(B) 210
(C) 84
(D) 35
(E) 7

Are You Up For the Challenge: 700 Level Questions

Since only two colors has to be used out of three, number of ways to select a combination of two colors out of three = Number of ways to select a color out of three * number of ways to select a color out of remaining two color.
= 3 * 2 = 6

Now, ways to arrange 7 stripes = 7!
Ways in which the two color(whatever they may be) stripes are arranged = 4!*3!
Ways to arrange 7 stripes with 2 colors = \(\frac{7!}{4!*3!}\)

Total number of ways to arrange 7 stripes with 2 colors = \(\frac{7!}{4!*3!}*6 = 210\)
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memayank
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Re: John has to paint a wall with seven horizontal stripes. He only has en [#permalink] New post   02 Sep 2021, 09:51
I have an issue. Where in the question it is written that he has to use 4+3 combination to paint the stripes. He may use 6+1, 5+2, 4+3, 3+4, 5+2, 6+1 Combinations also. And accordingly combinations will increase. Please suggest if I am wrong somewhere.
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Re: John has to paint a wall with seven horizontal stripes. He only has en [#permalink] New post   03 Sep 2021, 21:26
memayank wrote:
I have an issue. Where in the question it is written that he has to use 4+3 combination to paint the stripes. He may use 6+1, 5+2, 4+3, 3+4, 5+2, 6+1 Combinations also. And accordingly combinations will increase. Please suggest if I am wrong somewhere.

memayank
Read the question carefully and you would know where you are going here.
Quote:
John has to paint a wall with seven horizontal stripes. He only has enough paint for four red stripes, four blue stripes, and four yellow stripes. If he can use at most two colors, how many different ways can he paint the wall?

(A) 420
(B) 210
(C) 84
(D) 35
(E) 7

The max. stripes a color can have is 4. So, how come '6 + 1' or '5 + 2' OR vice-a-versa is possible.
Rest i guess you know how to solve.

Thanks..!!
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Re: John has to paint a wall with seven horizontal stripes. He only has en [#permalink] New post   16 Oct 2022, 10:52
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Re: John has to paint a wall with seven horizontal stripes. He only has en [#permalink]
16 Oct 2022, 10:52
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