Question Number 135861 by liberty last updated on 16/Mar/21
$${\color{mathred}{P}\color{mathred}{r}\color{mathred}{o}\color{mathred}{b}\color{mathred}{a}\color{mathred}{b}\color{mathred}{i}\color{mathred}{l}\color{mathred}{i}\color{mathred}{t}\color{mathred}{y}} \\ $$ Digits from 1 to 9 are randomly placed to form a 9-digit number. What is the probability that the number is divisible by 18?\\n
Answered by EDWIN88 last updated on 16/Mar/21
$$\mathrm{To}\:\mathrm{be}\:\mathrm{divisible}\:\mathrm{by}\:\mathrm{18},\:\mathrm{the}\:\mathrm{number}\:\mathrm{must}\:\mathrm{be} \\ $$ $$\mathrm{divisible}\:\mathrm{by}\:\mathrm{2}\:\mathrm{and}\:\mathrm{9}.\:\mathrm{The}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{digits}\:\mathrm{is} \\ $$ $$\mathrm{1}+\mathrm{2}+\mathrm{3}+\mathrm{4}+...+\mathrm{9}\:=\:\frac{\mathrm{9}×\mathrm{10}}{\mathrm{2}}\:=\:\mathrm{45}\:\mathrm{which}\:\mathrm{is}\: \\ $$ $$\mathrm{divisible}\:\mathrm{by}\:\mathrm{9},\:\mathrm{There}\:\mathrm{are}\:\mathrm{9}\:\mathrm{digits}\:\mathrm{of}\:\mathrm{which}\:\mathrm{are} \\ $$ $$\mathrm{even}\:\left(\mathrm{2},\mathrm{4},\mathrm{6},\mathrm{8}\right)\:.\:\mathrm{So}\:\mathrm{the}\:\mathrm{probability}\:\mathrm{that}\:\mathrm{the} \\ $$ $$\mathrm{number}\:\mathrm{is}\:\mathrm{divisible}\:\mathrm{by}\:\mathrm{18}\:\mathrm{is}\:\frac{\mathrm{4}}{\mathrm{9}} \\ $$