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Question
Three numbers are chosen from 1 to 20. Find the probability that they are not consecutive ______.
Options
`186/190`
`187/190`
`188/190`
`18/(""^20C_3)`
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Solution
Three numbers are chosen from 1 to 20. Find the probability that they are not consecutive `187/190`.
Explanation:
Set of three consecutive numbers from 1 to 20 are 1, 2, 3; 2, 3, 4; 3, 4, 5; ..., 18, 19, 20.
So, the probability that the numbers are consecutive
= `18/(""^20C_3)`
= `18/((201)/(3!17!))`
= `(18*3!*17!)/(20!)`
= `(18 xx 3 xx 2 xx 17!)/(20 xx 19 xx 18 xx 17!)`
= `(3 xx 2)/(20 xx 19)`
= `3/190`
∴ P(three numbers are not consecutive) = `1 - 3/190 = 187/190`
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