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# Richard has to paint a mural with seven horizontal stripes. He only ha

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Math Expert
Joined: 02 Sep 2009
Posts: 60625
Richard has to paint a mural with seven horizontal stripes. He only ha  [#permalink]

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06 Nov 2019, 06:08
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Difficulty:

75% (hard)

Question Stats:

53% (02:45) correct 47% (02:53) wrong based on 33 sessions

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Richard has to paint a mural with seven horizontal stripes. He only has enough paint for four red stripes, four blue stripes, four white stripes, four black stripes, and four yellow stripes. If his patron wants at most two different colors in the mural, how many different ways can he paint the wall?

(A) 120
(B) 350
(C) 700
(D) 2,520
(E) 5,040

Are You Up For the Challenge: 700 Level Questions

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Location: India
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Re: Richard has to paint a mural with seven horizontal stripes. He only ha  [#permalink]

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06 Nov 2019, 08:29
Bunuel wrote:
Richard has to paint a mural with seven horizontal stripes. He only has enough paint for four red stripes, four blue stripes, four white stripes, four black stripes, and four yellow stripes. If his patron wants at most two different colors in the mural, how many different ways can he paint the wall?

(A) 120
(B) 350
(C) 700
(D) 2,520
(E) 5,040

Are You Up For the Challenge: 700 Level Questions

total color options 5 out of which 2 are to be chosen so we have 5c2 ; 10 ways
and for 7 color stripes; they can be chosen in 7!/4!*3! * 2c1 = 70 ways
total possible colors = 70*10 ; 700
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Joined: 19 Oct 2018
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Re: Richard has to paint a mural with seven horizontal stripes. He only ha  [#permalink]

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06 Nov 2019, 08:34
The only possible combination for 7 horizontal stripes= 4 stripes of 1 color+ 3 strips of another color

So we need 2 colors; number of ways to select 2 color=5C2=10

Total possible arrangements= $$10*\frac{7!}{4!*3!}*2!$$=700

Bunuel wrote:
Richard has to paint a mural with seven horizontal stripes. He only has enough paint for four red stripes, four blue stripes, four white stripes, four black stripes, and four yellow stripes. If his patron wants at most two different colors in the mural, how many different ways can he paint the wall?

(A) 120
(B) 350
(C) 700
(D) 2,520
(E) 5,040

Are You Up For the Challenge: 700 Level Questions
Intern
Joined: 18 Aug 2018
Posts: 7
Re: Richard has to paint a mural with seven horizontal stripes. He only ha  [#permalink]

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06 Nov 2019, 10:25
1
The answer will be 700 as 7C4 for selecting stipes 5C1 for the first color and 4C1 for the second color.
so together 7C4 X 5C1X 4C1=700
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Re: Richard has to paint a mural with seven horizontal stripes. He only ha  [#permalink]

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07 Nov 2019, 19:56
Bunuel wrote:
Richard has to paint a mural with seven horizontal stripes. He only has enough paint for four red stripes, four blue stripes, four white stripes, four black stripes, and four yellow stripes. If his patron wants at most two different colors in the mural, how many different ways can he paint the wall?

(A) 120
(B) 350
(C) 700
(D) 2,520
(E) 5,040

Are You Up For the Challenge: 700 Level Questions

Since Richard needs to paint 7 stripes, he must use at least two colors since each color of paint is enough for only 4 stripes. However, since his patron wants at most two colors, he must use exactly two colors. The number of ways he can use 2 colors from 5 available colors is 5C2 = 10. For the pair of colors he chooses, he can paint 4 stripes of one color and 3 stripes of the other or vice versa. For example, if he chooses red and blue, he can paint 4 red stripes and 3 blue stripes or 3 red stripes and 4 blue stripes. Therefore, for a pair of colors, he has 2 choices to paint by the number of stripes of each color from the pair. Finally, once he has picked 2 colors and the number of stripes he’s going to paint with each color, there are 7!/(4! x 3!) = (7 x 6 x 5 x 4!)/(4! x 3 x 2) = 7 x 5 = 35 ways to paint the wall with the same 2 colors and the same number of stripes by each color. Therefore, in total, there are 10 x 2 x 35 = 700 different ways to paint the wall.

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Re: Richard has to paint a mural with seven horizontal stripes. He only ha   [#permalink] 07 Nov 2019, 19:56
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