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A box contains 50 tokens numbered 1 to 50. One token is picked at random. What is the probability that the number on the token is a multiple of 6 or 9?
11/50
12/50
9/50
13/50
11/50
Count multiples of 6 (50/6 = 8) and 9 (50/9 = 5), then subtract common multiples of LCM(6, 9)=18 (50/18 = 2). Result: (8 + 5 - 2)/50 = 11/50.
A set of tokens numbered from 1 to 50. One token is drawn at random.
P(AтИкB)=P(A)+P(B)тИТP(AтИйB)
Count multiples of 6 (50/6 = 8) and 9 (50/9 = 5), then subtract common multiples of LCM(6, 9)=18 (50/18 = 2). Result: (8 + 5 - 2)/50 = 11/50.
Students often double-count the multiples of 18 (i.e., 18 and 36) by simply adding the counts of multiples of 6 and 9 without using the inclusion-exclusion principle.
Find multiples of 6
Identify the count of multiples of 6 up to 50 by calculating тМК50/6тМЛ=8. These are {6, 12, 18, 24, 30, 36, 42, 48}.
n(A)=8
Find multiples of 9
Identify the count of multiples of 9 up to 50 by calculating тМК50/9тМЛ=5. These are {9, 18, 27, 36, 45}.
n(B)=5
Find common multiples
Find multiples of both 6 and 9 by finding multiples of their LCM, which is 18. These are {18, 36}.
n(AтИйB)=тМК50/18тМЛ=2
Apply inclusion-exclusion
Calculate the total favorable outcomes: 8+5тИТ2=11. The total number of tokens is 50.
P=508+5тИТ2тАЛ=5011тАЛ
A is correct because the application of the inclusion-exclusion principle results in 11 favorable tokens out of 50 total tokens.
This concept of overlapping sets is fundamental for solving Venn diagram problems in Set Theory and Logical Reasoning sections of competitive exams.