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Question Number 88876 by Joel578 last updated on 13/Apr/20

$$\mathrm{Given}\mathrm{a}10-\mathrm{digit}\mathrm{number}X=1345789026$$$$\mathrm{How}\mathrm{many}10-\mathrm{digit}\mathrm{number}\mathrm{that}\mathrm{can}\mathrm{be}\mathrm{made}$$$$\mathrm{using}\mathrm{every}\mathrm{digit}\mathrm{from}X,\mathrm{with}\mathrm{condition}:$$$$\mathrm{If}\mathrm{a}\mathrm{number}n\mathrm{is}\mathrm{located}\mathrm{in}{k}^{th}\mathrm{position}\mathrm{of}X,\mathrm{then}$$$$\mathrm{the}\mathrm{new}\mathrm{created}\mathrm{number}\mathrm{must}\mathrm{not}\mathrm{contain}$$$$\mathrm{number}n\mathrm{in}{k}^{th}\mathrm{position}$$$$$$$$\mathrm{Example}:$$$$\bullet \mathrm{Number}1\mathrm{is}\mathrm{located}\mathrm{in}{1}^{st}\mathrm{position}\mathrm{of}X,\mathrm{hence}$$$$1234567890\mathrm{is}\mathrm{not}\mathrm{valid},\mathrm{but}2134567890$$$$\mathrm{is}\mathrm{valid}$$$$\bullet \mathrm{Number}5\mathrm{and}0\mathrm{are}\mathrm{located}\mathrm{in}{4}^{th}\mathrm{and}{8}^{th}\mathrm{position}$$$$\mathrm{of}X,\mathrm{hence}9435162087\mathrm{is}\mathrm{not}\mathrm{valid},\mathrm{but}$$$$9431506287\mathrm{is}\mathrm{valid}.$$$$\bullet 1345026789\mathrm{is}\mathrm{not}\mathrm{valid}$$$$\bullet \mathrm{and}\mathrm{so}\mathrm{on}...$$

Commented by Joel578 last updated on 13/Apr/20

$$\mathrm{I}\mathrm{got}:$$$$\left(\begin{array}{c}10\\ 0\end{array}\right){10}^{10}-\left(\begin{array}{c}10\\ 1\end{array}\right)9^9+\left(\begin{array}{c}10\\ 2\end{array}\right)8^8-...+\left(\begin{array}{c}10\\ 10\end{array}\right)1$$$$\mathrm{is}\mathrm{it}\mathrm{correct}?$$

Commented by mr W last updated on 29/Mar/21

$$\frac{8}{9}\times !10=1186632$$

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