document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=252429>Question 252429</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Principles of Counting or Permutations.  What formula do I use here.\r<BR>\n" );
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document.write( "Telephone Numbers.  How many 7-digit telephone numbers are possible if the first digit cannot be 0 and:<BR>\n" );
document.write( "a.)  Only odd digits may be used.  Answer: (78,125)<BR>\n" );
document.write( "b.)  The telephone number must be a multiple of 10 (that is, it must end in 0).  Answer:(900,000)<BR>\n" );
document.write( "c.)  The telephone number must be a multiple of 100. Answer:(90,000)<BR>\n" );
document.write( "d.)  The first 3 digits are 481. Answer:(10,000)<BR>\n" );
document.write( "e.)  No repetitions are allowed. Answer:(544,320)<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #184333 by </B><A HREF=/tutors/aboutme.mpl?userid=palanisamy><B>palanisamy(496)</B></A><img src=\"/images/stars/stars-10.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=palanisamy><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=184333\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> We have to form 7 digit telephone numbers.<BR>\n" );
document.write( "a)The odd numbers are 1,3,5,7,9<BR>\n" );
document.write( "  Every digit can be filled with any one of these numbers in 5 ways.<BR>\n" );
document.write( "  So the number of 7 digit numbers = 5*5*5*5*5*5*5<BR>\n" );
document.write( "                                   = 78,125<BR>\n" );
document.write( "b)The telephone number must be a multiple of 10 <BR>\n" );
document.write( "  So, the noumber must end with 0. Fill the unit place with 0.<BR>\n" );
document.write( "  This can be done in only one way. <BR>\n" );
document.write( "  The number cannot begin with 0.<BR>\n" );
document.write( "  So the first place can be filled with any one of the numbers   <BR>\n" );
document.write( "  1,2,3,4,5,6,7,8,9 <BR>\n" );
document.write( "  This can be done in 9 ways.<BR>\n" );
document.write( "  The second place can be filled with any one of the numbers <BR>\n" );
document.write( "  0,1,2,3,4,5,6,7,8,9<BR>\n" );
document.write( "  This can be done in 10 ways.<BR>\n" );
document.write( "  The third place can be filled with any one of the numbers <BR>\n" );
document.write( "  0,1,2,3,4,5,6,7,8,9<BR>\n" );
document.write( "  This can be done in 10 ways.<BR>\n" );
document.write( "  The fourth place can be filled with any one of the numbers <BR>\n" );
document.write( "  0,1,2,3,4,5,6,7,8,9<BR>\n" );
document.write( "  This can be done in 10 ways.<BR>\n" );
document.write( "  The fifth place can be filled with any one of the numbers <BR>\n" );
document.write( "  0,1,2,3,4,5,6,7,8,9<BR>\n" );
document.write( "  This can be done in 10 ways.<BR>\n" );
document.write( "  The sixth place can be filled with any one of the numbers <BR>\n" );
document.write( "  0,1,2,3,4,5,6,7,8,9<BR>\n" );
document.write( "  This can be done in 10 ways.<BR>\n" );
document.write( "  Therefore the total number of 7 digit telephone numbers <BR>\n" );
document.write( "  which are multiples of 10 = 1*9*10*10*10*10*10<BR>\n" );
document.write( "                            = 900,000\r<BR>\n" );
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document.write( "c)The telephone number must be a multiple of 100 <BR>\n" );
document.write( "  Fill the unit place with 0 and the tenth place with 0.<BR>\n" );
document.write( "  This can be done in only one way. <BR>\n" );
document.write( "  The number cannot begin with 0.<BR>\n" );
document.write( "  So the first place can be filled with any one of the numbers   <BR>\n" );
document.write( "  1,2,3,4,5,6,7,8,9 <BR>\n" );
document.write( "  This can be done in 9 ways.<BR>\n" );
document.write( "  The second place can be filled with any one of the numbers <BR>\n" );
document.write( "  0,1,2,3,4,5,6,7,8,9<BR>\n" );
document.write( "  This can be done in 10 ways.<BR>\n" );
document.write( "  The third place can be filled with any one of the numbers <BR>\n" );
document.write( "  0,1,2,3,4,5,6,7,8,9<BR>\n" );
document.write( "  This can be done in 10 ways.<BR>\n" );
document.write( "  The fourth place can be filled with any one of the numbers <BR>\n" );
document.write( "  0,1,2,3,4,5,6,7,8,9<BR>\n" );
document.write( "  This can be done in 10 ways.<BR>\n" );
document.write( "  The fifth place can be filled with any one of the numbers <BR>\n" );
document.write( "  0,1,2,3,4,5,6,7,8,9<BR>\n" );
document.write( "  This can be done in 10 ways.<BR>\n" );
document.write( "  Therefore the total number of 7 digit telephone numbers <BR>\n" );
document.write( "  which are multiples of 100 = 1*9*10*10*10*10*<BR>\n" );
document.write( "                            = 90,000\r<BR>\n" );
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document.write( "d)Fill the first three digits as 481.<BR>\n" );
document.write( "  This can be done in only one way. <BR>\n" );
document.write( "  The fourth place can be filled with any one of the numbers <BR>\n" );
document.write( "  0,1,2,3,4,5,6,7,8,9<BR>\n" );
document.write( "  This can be done in 10 ways.<BR>\n" );
document.write( "  The fifth place can be filled with any one of the numbers <BR>\n" );
document.write( "  0,1,2,3,4,5,6,7,8,9<BR>\n" );
document.write( "  This can be done in 10 ways.<BR>\n" );
document.write( "  The sixth place can be filled with any one of the numbers <BR>\n" );
document.write( "  0,1,2,3,4,5,6,7,8,9<BR>\n" );
document.write( "  This can be done in 10 ways.<BR>\n" );
document.write( "  The seventh place can be filled with any one of the numbers <BR>\n" );
document.write( "  0,1,2,3,4,5,6,7,8,9<BR>\n" );
document.write( "  This can be done in 10 ways.<BR>\n" );
document.write( "  Therefore the total number of 7 digit telephone numbers <BR>\n" );
document.write( "  which begin with 481 are  = 10*10*10*10*<BR>\n" );
document.write( "                            = 10,000\r<BR>\n" );
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document.write( "e) No repetitions are allowed.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  The number cannot begin with 0.<BR>\n" );
document.write( "  So the first place can be filled with any one of the numbers   <BR>\n" );
document.write( "  1,2,3,4,5,6,7,8,9 <BR>\n" );
document.write( "  This can be done in 9 ways.<BR>\n" );
document.write( "  After filling the first place we have 9 numbers and so<BR>\n" );
document.write( "  the second place can be filled with any one of the numbersin 9 ways. <BR>\n" );
document.write( "  After filling the second place we have 8 numbers and so<BR>\n" );
document.write( "  the third place can be filled with any one of the numbersin 8 ways.<BR>\n" );
document.write( "  After filling the third place we have 7 numbers and so<BR>\n" );
document.write( "  the foutrh place can be filled with any one of the numbersin 7 ways.<BR>\n" );
document.write( "  After filling the fourth place we have 6 numbers and so<BR>\n" );
document.write( "  the fifth place can be filled with any one of the numbersin 6 ways.<BR>\n" );
document.write( "  After filling the fifth place we have 5 numbers and so<BR>\n" );
document.write( "  the sixth place can be filled with any one of the numbersin 5 ways. <BR>\n" );
document.write( "  After filling the sixth place we have 4 numbers and so<BR>\n" );
document.write( "  the seventh place can be filled with any one of the numbersin 4 ways. <BR>\n" );
document.write( "  Therefore the total number of 7 digit telephone numbers <BR>\n" );
document.write( "  which are formed without repetition                                =9*9*8*7*6*5*4                                      <BR>\n" );
document.write( "= 544,320\r<BR>\n" );
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document.write( " </SPAN>\n" );
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