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Question Number 18464 by chernoaguero@gmail.com last updated on 22/Jul/17

$$3\mathrm{numbers}\mathrm{are}\mathrm{chosen}\mathrm{from}1\mathrm{to}30.\mathrm{The}$$$$\mathrm{probability}\mathrm{that}\mathrm{they}\mathrm{are}\mathrm{not}\mathrm{consecutive}$$$$\mathrm{is}$$

Answered by Tinkutara last updated on 22/Jul/17

$$\mathrm{Total}\mathrm{number}\mathrm{of}\mathrm{ways}\mathrm{in}\mathrm{which}\mathrm{a}\mathrm{triplet}$$$$\mathrm{of}3\mathrm{numbers}\mathrm{can}\mathrm{be}\mathrm{chosen}\mathrm{is}{}^{30}{C}_3.$$$$\mathrm{Now}\mathrm{we}\mathrm{have}28\mathrm{pairs}\mathrm{of}3\mathrm{consecutive}$$$$\mathrm{integers},\mathrm{i}.\mathrm{e}.(1,2,3),(2,3,4),(3,4,5),$$$$...,(28,29,30).$$$$\mathrm{Probability}\mathrm{of}\mathrm{these}\mathrm{consecutive}\mathrm{integers}$$$$=\frac{28}{{}^{30}{C}_3}=\frac{28}{4060}=\frac{1}{145}$$$$\therefore \mathrm{Probability}\mathrm{that}\mathrm{they}\mathrm{are}\mathrm{not}$$$$\mathrm{consecutive}\mathrm{integers}\mathrm{is}=\frac{144}{145}$$

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