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There are five identical green dyes, four identical blue dyes, three identical yellow dyes and four different red dyes. How many combinations of dyes can be chosen taking at least one green, at least one blue and at least one red dye?
- a)980
- b)1024
- c)1200
- d)320
Correct answer is option 'C'. Can you explain this answer?
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There are five identical green dyes, four identical blue dyes, three ...
For Green dyes: Since at least one green dye has to be selected, we can do so in 5 ways by selecting either 1,2, 3, 4 or all 5 green dyes. Since the dyes are identical, it does not matter which dye is selected. Thus, only the count of dyes leads to possibilities.
For Blue dyes: 4 ways (Similar calculation as in the case of Green dyes) For Yellow dyes: In case of yellow dyes, there is no ‘atleast’ constraint, thus we can select yellow dyes in 3 + 1 = 4 ways by selecting either 0, 1, 2 or all 3 dyes. This means that we might have a selection in which there is not even one yellow dye.
For Red dyes: Red dyes are all different from each other. Thus in this case, it matters which dye is selected and also how many dyes are selected. Thus, for every dye we have two options - either select it or leave it. Thus for each of the 4 red dyes we have 2 possibilities each, leading to a total of 2 x 2 x 2 x 2 = 24 ways.
But these 24 ways include a possibility in which none of the red dyes is selected. But we want at least one red dye in our selections. Hence we subtract this possibility from the total. 24 - 1 =15 ways The total number of ways = 5 x 4 x 4 x (24 - 1) = 1200 Hence, option 3.
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There are five identical green dyes, four identical blue dyes, three identical yellow dyes and four different red dyes. How many combinations of dyes can be chosen taking at least one green, at least one blue and at least one red dye?a)980b)1024c)1200d)320Correct answer is option 'C'. Can you explain this answer?
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