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
IMO C

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

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

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.

Answer: C
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At most 2 colours means that we can use either one or only two colours. Since one colour is not enough as each colour is only sufficient to paint 4 stripes, we have to use 2 colors. The only possibility is 4 stripes of one colour and 3 stripes of another colour. Now ,we have 5 colours to chose for the first 4 stripes, hence 5 ways. Once chosen, we have 4 colours to chose for 3 stripes, hence 4 ways. So we can choose colours in 5 x 4 = 20 ways such that each colour represents 4 and 3 stripes.one of the arrangement will look like this BBBBGGG. Now just think about it , these 7 stripes can itself be arranged. It can be BGGBGBB or other combinations. So these 7 stripes can be arranged in 7C4 ways which will be 35 ways. Thus total arrangements will be 35 x 20 = 700 ways. Hence C is the answer
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