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Suppose an integer from 1 through 1000 is chosen at random, find the probability that the integer is a multiple of 2 or a multiple of 9. Select the correct answer from above options

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Multiple of 2 from 1 to 1000 are 2, 4, 6, 8,…., 1000 Let n be the number of terms of above series. ∴ nth term = 1000 ∴ ⇒ 2 + ( n − 1 ) 2 = 1000 ⇒ ⇒ 2 + 2 n − 2 = 1000 ⇒ ⇒ 2 n = 1000 ⇒ ∴ n = 500 ∴ Since, the number of multiple of 2 are 500. So, the multiple of 9 are 9, 18, 27,....,999 Let m be the number of term in above series. ∴ mth term = 999 ∴ ⇒ 9 + ( m − 1 ) 9 = 999 ⇒ ⇒ 9 + 9 m − 9 = 999 ⇒ ⇒ 9 m = 999 ⇒ ∴ m = 111 ∴ Since, the number of multiple of 9 are 111. So, the multiple of 2 and 9 both are 18, 36,..., 990 Let p be the number of terms in above series. ∴ pth term = 990 ∴ ⇒ 18 + ( p − 1 ) 18 = 990 ⇒ ⇒ 18 + 18 p − 18 = 990 ⇒ ⇒ 18 p = 990 ⇒ ∴ p = 990 18 = 55 ∴ Since, the number of multiple of 2 and 9 are 55. ∴ Number of miltiple of 2 or 9 = 500 + 111-55 = 556 ∴ ∴ Required probability = n ( E ) n ( S ) = 556 1000 = 0.556 ∴

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