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A bag contains balls numbered from 1 to 42. One ball is drawn at random from these balls. The probability that its number is a multiple of 7 or 8 is:
4222тАЛ
2213тАЛ
2211тАЛ
4211тАЛ
4211тАЛ
Count multiples of 7 (6 balls) and 8 (5 balls). The LCM of 7 and 8 is 56, which is > 42, so there is no intersection (0 overlaps). Total = 6 + 5 = 11.
A set of balls numbered from 1 to 42. Total outcomes = 42.
P(AтИкB)=P(A)+P(B)тИТP(AтИйB)
Count multiples of 7 (6 balls) and 8 (5 balls). The LCM of 7 and 8 is 56, which is > 42, so there is no intersection (0 overlaps). Total = 6 + 5 = 11.
Students often try to find an intersection (like 56) when there is none within the range, leading to incorrect subtraction or double counting.
Identify multiples of 7
Multiples of 7 up to 42 are: 7, 14, 21, 28, 35, 42. There are 6 such numbers.
n(A)=6
Identify multiples of 8
Multiples of 8 up to 42 are: 8, 16, 24, 32, 40. There are 5 such numbers.
n(B)=5
Check for intersection
The LCM of 7 and 8 is 56. Since 56 > 42, there are no common multiples in the range.
n(AтИйB)=0
Calculate probability
Total successful outcomes are 6 + 5 = 11. The probability is the number of successful outcomes divided by total balls (42).
P=4211тАЛ
D is correct because the total number of multiples of 7 or 8 within the range of 42 is 11, resulting in a probability of 11/42.
This type of inclusion-exclusion principle is common in number theory and counting problems; always check for the LCM of the denominators to see if there is an overlap.
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