6.3 # 22 a. How many integers from 1 through 1,000 are multiples of 2or multiple

Question

6.3 # 22 a. How many integers from 1 through 1,000 are multiples of 2or multiples of 9? b. Suppose an integer from 1 through 1,000 is choosen atrandom. Use the reult of part (a) to find the probability that theinteger is amultiple of 2 or a multiple of 9. c. How many integers from 1 through 1,000 are neithermultiples of 2 nor multiples of 9? (Please I need explaination step by step to help me understandit) 6.3 # 22 a. How many integers from 1 through 1,000 are multiples of 2or multiples of 9? b. Suppose an integer from 1 through 1,000 is choosen atrandom. Use the reult of part (a) to find the probability that theinteger is amultiple of 2 or a multiple of 9. c. How many integers from 1 through 1,000 are neithermultiples of 2 nor multiples of 9? (Please I need explaination step by step to help me understandit)

Explanation / Answer

Say we call: T the set of numbers divisible by 2 (up to 1000) N the set of numbers divisible by 9 (up to 1000) E the set of numbers divisible by both 2 and 9 (up to 1000) -- notethat this set is the intersection of T and N, the numbers divisibleby 18 Then we know: T contains 500 elements -- |T| = 500 N contains 111 elements -- |N| = 111 E contains 55 elements -- |E| = 55 a) Using the Inclusion-Exclusion principle: |T| + |N| - |E| = 500 + 111 - 55 = 556 b) Out of a 1000 numbers, 556 are a multiple of 2 or 9 Resulting in the probability: 556/1000 = 0.556 c) The ones we didn't count in a) 1000 - 556 = 444

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