The numbers 1,2,3, .. , n are arranged in a random order. The probability that the digits 1,2,3,..,k appear as neighbour is ?

Asked by Paavani | 26th Dec, 2015, 10:51: AM

Expert Answer:

bold Total bold space bold number bold space bold of bold space bold ways bold space bold of bold space bold arranging bold space bold apostrophe bold n bold apostrophe bold space bold distinct bold space bold numbers bold equals bold n bold factorial bold. bold. bold. bold. bold. bold left parenthesis bold 1 bold right parenthesis bold If bold space bold apostrophe bold k bold apostrophe bold space bold of bold space bold these bold space bold numbers bold space bold are bold space bold neighbours bold space bold always bold comma bold space bold we bold space bold can bold space bold consider bold space bold these bold space bold apostrophe bold k bold apostrophe bold space bold things bold space bold as bold space bold one bold space bold unit bold. stack bold 1 bold comma bold space bold 2 bold comma bold space bold 3 bold comma bold. bold. bold. bold. bold comma bold space bold k with bold underbrace below bold comma bold space bold k bold plus bold 1 bold comma bold space bold K bold plus bold 2 bold comma bold. bold. bold. bold comma bold space bold n bold So bold space bold effectively bold space bold we bold space bold are bold space bold making bold space bold arrangement bold space bold of bold space bold n bold minus bold k bold plus bold 1 bold space bold numbers bold. bold Total bold space bold number bold space bold of bold space bold arrangement bold space bold possible bold space bold for bold space bold this bold space bold is bold space open parentheses bold n bold minus bold k bold plus bold 1 close parentheses bold factorial bold apostrophe bold k bold apostrophe bold space bold things bold space bold can bold space bold also bold space bold be bold space bold arranged bold space bold among bold space bold themselves bold space bold in bold space bold k bold factorial bold space bold ways bold. bold So bold comma bold space bold favorable bold space bold number bold space bold of bold space bold ways bold equals open parentheses bold n bold minus bold k bold plus bold 1 close parentheses bold factorial bold cross times bold k bold factorial bold So bold space bold probability bold space bold equals bold space fraction numerator open parentheses bold n bold minus bold k bold plus bold 1 close parentheses bold factorial bold cross times bold k bold factorial over denominator bold n bold factorial end fraction

Answered by satyajit samal | 28th Dec, 2015, 04:59: PM

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