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26 Nov 2020, 23:09
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Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
75%
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Question Stats:
55% (02:28) correct
45%
(02:37)
wrong
based on 153
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There are 5 different green dyes, 4 different blue dyes and 3 different red dyes. How many combinations of dyes can be chosen taking at least 1 green and at least 1 blue dye in each combination.
A. 36
B. 60
C. 1533
D. 3720
E. 4096
A. 36
B. 60
C. 1533
D. 3720
E. 4096
26 Nov 2020, 23:18
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Explanation:
No. of ways to select atleast 1 green dye = 5C1 + 5C2 + 5C3 + 5C4 + 5C5 = 2^5 - 1
Similarly , No. of ways to select atleast 1 blue dye = 2^4 - 1
Any no. red dye can be chosen = 3C0 + 3C1 + 3C2 + 3C3 = 2^3
total ways=(2^5 - 1)*(2^4 -1)*(2^3)
= 31 × 15 × 8
= 3720
IMO-D
No. of ways to select atleast 1 green dye = 5C1 + 5C2 + 5C3 + 5C4 + 5C5 = 2^5 - 1
Similarly , No. of ways to select atleast 1 blue dye = 2^4 - 1
Any no. red dye can be chosen = 3C0 + 3C1 + 3C2 + 3C3 = 2^3
total ways=(2^5 - 1)*(2^4 -1)*(2^3)
= 31 × 15 × 8
= 3720
IMO-D
27 Nov 2020, 08:59
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A different way: with no restrictions at all, we'd have 2 choices for each dye, use it in or combination or do not use it. So with no restrictions, there would be 2^12 possibilities (including choosing no dyes at all). Almost all of those combinations will use at least one green and one blue dye, so the answer will not be much smaller than 2^12 = 4096, and only one answer makes sense.
If we did want an exact answer: we don't want to count possibilities if they use only green and red dyes, and (ignoring the blue dyes altogether) there are 2^8 of those. We also don't want to count the 2^7 possibilities which avoid green dyes altogether. But 2^8 + 2^7 overcounts the number of combinations we'd like to avoid, because in both counts, we counted the 2^3 combinations that use exclusively red dye with no green or blue at all. So there are 2^8 + 2^7 - 2^3 combinations we do not want to count, and the exact answer is
2^12 - (2^8 + 2^7 - 2^3) = 2^12 - 2^8 - 2^7 + 2^3 = 2^3 (2^9 - 2^5 - 2^4 + 1) = 2^3 (512 - 32 - 16 + 1) = (8)(465) = 3720
though with answers this far apart, estimation is much faster than exact calculation.
If we did want an exact answer: we don't want to count possibilities if they use only green and red dyes, and (ignoring the blue dyes altogether) there are 2^8 of those. We also don't want to count the 2^7 possibilities which avoid green dyes altogether. But 2^8 + 2^7 overcounts the number of combinations we'd like to avoid, because in both counts, we counted the 2^3 combinations that use exclusively red dye with no green or blue at all. So there are 2^8 + 2^7 - 2^3 combinations we do not want to count, and the exact answer is
2^12 - (2^8 + 2^7 - 2^3) = 2^12 - 2^8 - 2^7 + 2^3 = 2^3 (2^9 - 2^5 - 2^4 + 1) = 2^3 (512 - 32 - 16 + 1) = (8)(465) = 3720
though with answers this far apart, estimation is much faster than exact calculation.
