document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=766577>Question 766577</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The number of five digit telephone numbers having atleast one of their digits repeated is<BR>\n" );
document.write( "a)900000 b)100000 c)30240 d)65760 </SPAN>\n" );
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document.write( "Please check - doesn't look like any of the 4 choices is correct.\r<BR>\n" );
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document.write( "Here is the solution:\r<BR>\n" );
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document.write( "How many 5 digit telephone numbers are there in total?<BR>\n" );
document.write( "From 10000 through 99999, there are 90000 of them.<BR>\n" );
document.write( "Let us see - how many have all 5 digits different?<BR>\n" );
document.write( "If all 5 have to be different,<BR>\n" );
document.write( "1) The first digit can be filled in 9 ways (it cannot be 0)<BR>\n" );
document.write( "2) Second can be filled in 9 ways (since 0 can be the second digit)<BR>\n" );
document.write( "3) 3rd can be filled 8 ways, 4th in 7 ways and 5th in 6 ways. (Because we cannot reuse a digit)\r<BR>\n" );
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document.write( "So total number of 5 digit phone numbers with all the digits different is<BR>\n" );
document.write( "9*9*8*7*6 = 27216.\r<BR>\n" );
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document.write( "Now, it is clear that all the other 5 digit phone numbers must have at least 1 digit repeating, since only 27216 of them have all the 5 digits as different!\r<BR>\n" );
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document.write( "So, the number of 5 digit phone numbers with at least 1 repeating digit = 90000 - 27216 = 62784.\r<BR>\n" );
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document.write( ":) </SPAN>\n" );
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