document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1020312>Question 1020312</A>: <I><SPAN STYLE=\"word-break: break-all;\"> if a cat can climb 1 or 2 steps at a time. in how many ways can she climb a flight of 10 stairs\r<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 #636344 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><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/cgi-bin/show-face.mpl?user=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=636344\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> if we let a = the cat takes 1 step at a time, and we let b = the cat takes 2 steps at a time, then we will get the following possible combinations.\r<BR>\n" );
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document.write( "5b + 0a<BR>\n" );
document.write( "4b + 2a<BR>\n" );
document.write( "3b + 4a<BR>\n" );
document.write( "2b + 6a<BR>\n" );
document.write( "1b + 8a<BR>\n" );
document.write( "0b + 10a\r<BR>\n" );
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document.write( "5b means the cat takes 10 stair steps 2 at a time for a total of 5 cat steps.\r<BR>\n" );
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document.write( "4b + 2a means the cat takes 8 stair steps 2 at a time and 2 stair steps 1 at a time for a total of 6 cat steps.\r<BR>\n" );
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document.write( "order is important because the cat can take 1 stair step at a time and then 2 stair steps at a time, or the cat can take 2 stair steps at a time and then 1 stair step at a time.  each of those counts as a different possible way for the cat to climb the stair steps.\r<BR>\n" );
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document.write( "5b + 0a can occur in 5! / (5! * 0!) = 1 way.<BR>\n" );
document.write( "4b + 2a can occur in 6! / (4! * 2!) = 15 ways.<BR>\n" );
document.write( "3b + 4a can occur in 7! / (3! * 4!) = 35 ways.<BR>\n" );
document.write( "2b + 6a can occur in 8! / (2! * 6!) = 28 ways.<BR>\n" );
document.write( "1b + 8a can occur in 9! / (1! * 8!) = 9 ways<BR>\n" );
document.write( "0b + 10a can occur in 10! / (0! * 10!) = 1 way.\r<BR>\n" );
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document.write( "the total possible number of ways that the cat can climb the stair steps is therefore 1 + 15 + 35 + 28 + 9 + 1 = 89 ways.\r<BR>\n" );
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document.write( "we'll detail 4b + 2a so you can see how it works.\r<BR>\n" );
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document.write( "the possible ways are:\r<BR>\n" );
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document.write( "bbbbaa\r<BR>\n" );
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document.write( "bbbaab<BR>\n" );
document.write( "bbbaba\r<BR>\n" );
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document.write( "bbaabb<BR>\n" );
document.write( "bbabab<BR>\n" );
document.write( "bbabba\r<BR>\n" );
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document.write( "baabbb<BR>\n" );
document.write( "bababb<BR>\n" );
document.write( "babbab<BR>\n" );
document.write( "babbba\r<BR>\n" );
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document.write( "aabbbb<BR>\n" );
document.write( "ababbb<BR>\n" );
document.write( "abbabb<BR>\n" );
document.write( "abbbab<BR>\n" );
document.write( "abbbba\r<BR>\n" );
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document.write( "that's a total of 15 possible ways.\r<BR>\n" );
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document.write( "the others work the same way.\r<BR>\n" );
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document.write( "5b + 0a can only be bbbbb<BR>\n" );
document.write( "0b + 10a can only be aaaaaaaaaa\r<BR>\n" );
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document.write( "1b + 8a can be:\r<BR>\n" );
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document.write( "baaaaaaaa<BR>\n" );
document.write( "abaaaaaaa<BR>\n" );
document.write( "aabaaaaaa<BR>\n" );
document.write( "aaabaaaaa<BR>\n" );
document.write( "aaaabaaaa<BR>\n" );
document.write( "aaaaabaaa<BR>\n" );
document.write( "aaaaaabaa<BR>\n" );
document.write( "aaaaaaaba<BR>\n" );
document.write( "aaaaaaaab\r<BR>\n" );
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document.write( "that a total of 9 possible ways.\r<BR>\n" );
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document.write( "if i did this correctly, then the total possible number of ways will be 89.\r<BR>\n" );
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document.write( "i detailed all the possible ways if there were only 6 steps instead of 10 and the method i used looks to be correct.  therefore i'm assuming that the same method will work with 10 steps even though i didn't detail all the possible ways, but just some of them.\r<BR>\n" );
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document.write( "it is interesting that the combination formula will provide the same answer.\r<BR>\n" );
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document.write( "for example:\r<BR>\n" );
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document.write( "7! / (3! * 4!)= 35 and c(7,3) will also give you 35.\r<BR>\n" );
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document.write( "the combination formula is c(n,x) = n! / (x! * (n-x)!)\r<BR>\n" );
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document.write( "then n = 7 and x = 3, you get c(7,3) = 7! / (3! * 4!).\r<BR>\n" );
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document.write( "they're identical, so you can expect to get the same answer.\r<BR>\n" );
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document.write( "the key to solving this problem was not the formulas, but how to set up the problem logically.\r<BR>\n" );
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document.write( "that i did up front when i determined the possible combinations were:\r<BR>\n" );
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document.write( "5b<BR>\n" );
document.write( "4b + 2a<BR>\n" );
document.write( "etc.....\r<BR>\n" );
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document.write( "i used the bigger variable as the controlling variable.\r<BR>\n" );
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document.write( "with 1b, you can have 8 a's because 1b takes 2 steps and there are 8 steps remaining.\r<BR>\n" );
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document.write( "with 2b, you can have 6 a's because 2b's take 4 steps and there are 6 steps remaining.\r<BR>\n" );
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document.write( "etc.....\r<BR>\n" );
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