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Question Number 100971 by john santu last updated on 29/Jun/20

A woman sent 8 letters to her friends. The letters are kept in the addressed envelopes at random. The probability that 4 friends receive correct letters and 4 letters go to wrong destination, is ___

$$\mathrm{A}\:\mathrm{woman}\:\mathrm{sent}\:\mathrm{8}\:\mathrm{letters}\:\mathrm{to}\:\mathrm{her}\: \\ $$$$\mathrm{friends}.\:\mathrm{The}\:\mathrm{letters}\:\mathrm{are}\:\mathrm{kept}\:\mathrm{in}\:\mathrm{the} \\ $$$$\mathrm{addressed}\:\mathrm{envelopes}\:\mathrm{at}\:\mathrm{random}.\: \\ $$$$\mathrm{The}\:\mathrm{probability}\:\mathrm{that}\:\mathrm{4}\:\mathrm{friends}\: \\ $$$$\mathrm{receive}\:\mathrm{correct}\:\mathrm{letters}\:\mathrm{and}\:\mathrm{4}\:\mathrm{letters}\: \\ $$$$\mathrm{go}\:\mathrm{to}\:\mathrm{wrong}\:\mathrm{destination},\:\mathrm{is}\:\_\_\_\: \\ $$

Answered by 1549442205 last updated on 01/Jul/20

There are C_8 ^4 ways to choose set of four friends that woman send correctly.For set of four remaining friends there are 12 ways that woman send false (4 remaining letters didn′t go to true addresses)while there are a total 8! ways to put letters into 8 addressed envelopes.So,P=((12.C_8 ^4 )/(8!))=(1/(48))

$$\mathrm{There}\:\:\mathrm{are}\:\mathrm{C}_{\mathrm{8}} ^{\mathrm{4}} \:\mathrm{ways}\:\mathrm{to}\:\mathrm{choose}\:\mathrm{set}\:\mathrm{of}\:\mathrm{four}\: \\ $$$$\mathrm{friends}\:\mathrm{that}\:\mathrm{woman}\:\mathrm{send}\:\mathrm{correctly}.\mathrm{For} \\ $$$$\mathrm{set}\:\mathrm{of}\:\mathrm{four}\:\mathrm{remaining}\:\mathrm{friends}\:\mathrm{there}\:\mathrm{are}\: \\ $$$$\mathrm{12}\:\mathrm{ways}\:\mathrm{that}\:\mathrm{woman}\:\mathrm{send}\:\mathrm{false}\: \\ $$$$\left(\mathrm{4}\:\mathrm{remaining}\:\mathrm{letters}\:\mathrm{didn}'\mathrm{t}\:\mathrm{go}\:\mathrm{to}\:\mathrm{true}\:\right. \\ $$$$\left.\mathrm{addresses}\right)\mathrm{while}\:\mathrm{there}\:\mathrm{are}\:\mathrm{a}\:\mathrm{total}\:\mathrm{8}!\: \\ $$$$\mathrm{ways}\:\mathrm{to}\:\mathrm{put}\:\mathrm{letters}\:\mathrm{into}\:\mathrm{8}\:\: \\ $$$$\mathrm{addressed}\:\mathrm{envelopes}.\mathrm{So},\mathrm{P}=\frac{\mathrm{12}.\mathrm{C}_{\mathrm{8}} ^{\mathrm{4}} }{\mathrm{8}!}=\frac{\mathrm{1}}{\mathrm{48}} \\ $$$$ \\ $$

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