document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=965989>Question 965989</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Q.1 IF THE RATIO OF AM AND GM OF TWO NUMBER IS 5:4. THE RATIO BETWEEN THE NUMBERS IS 4:1? TRUE / FALSE.. Please advise\r<BR>\n" );
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document.write( "Q.2 1.	THE NUMBER OF PERMUTATION OF N OBJECTS TAEKN TOGETHER OF WHICH A OBJECTS ARE SAME TYPE AND B OBJECTS ARE OF SAME TYPE AND REMAINING ARE OF SAME TYPE IS n! / a!b!(n-a-b)! ?  TRUE / FALSE   Please advise\r<BR>\n" );
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document.write( "Thanks well in advance.<BR>\n" );
document.write( "Regards- Raheeq </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #590412 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=590412\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> \r<BR>\n" );
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document.write( "Q.1 IF THE RATIO OF AM AND GM OF TWO NUMBER IS 5:4. THE RATIO BETWEEN THE NUMBERS IS 4:1? TRUE / FALSE..\r<BR>\n" );
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document.write( "here's a definition of geometric mean:<BR>\n" );
document.write( "<a  rel=nofollow HREF= \"http://www.mathwords.com/g/geometric_mean.htm\" target = \"_blank\">http://www.mathwords.com/g/geometric_mean.htm</a>\r<BR>\n" );
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document.write( "here's a definition of arithmetic mean:<BR>\n" );
document.write( "<a  rel=nofollow HREF= \"http://www.mathwords.com/a/arithmetic_mean.htm\" target = \"_blank\">http://www.mathwords.com/a/arithmetic_mean.htm</a>\r<BR>\n" );
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document.write( "based on those definitions, you get the following;\r<BR>\n" );
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document.write( "AM of a and b is (a+b)/2\r<BR>\n" );
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document.write( "GM of a and b is sqrt(a*b)\r<BR>\n" );
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document.write( "the ratio of AM to GM is equal to AM/GM which is equal to 5/4.\r<BR>\n" );
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document.write( "this means that AM = 5 and Gm = 4, or any multiple of that.\r<BR>\n" );
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document.write( "for example, if AM = 10, then GM = 8.\r<BR>\n" );
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document.write( "the ratio is still 5/4.\r<BR>\n" );
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document.write( "we'll set AM = 5 and GM = 4 and solve using those values.\r<BR>\n" );
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document.write( "AM = (a+b)/2.\r<BR>\n" );
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document.write( "since AM = 5, this means that (a+b)/2 = 5.\r<BR>\n" );
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document.write( "solve for a to get a = 10 - b.\r<BR>\n" );
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document.write( "GM = sqrt(a*b).\r<BR>\n" );
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document.write( "since GM = 4, this means that sqrt(a*b) = 4\r<BR>\n" );
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document.write( "since a = 10 - b, then sqrt(a*b) = 4 becomes sqrt((10-b)*b) = 4.\r<BR>\n" );
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document.write( "square both sides of this equation to get (10-b)*b = 16\r<BR>\n" );
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document.write( "simplify to get 10b - b^2 = 16\r<BR>\n" );
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document.write( "add b^2 to both sides of the equation and subtract 10b from both sides of the equation to get 0 = b^2 - 10b + 16\r<BR>\n" );
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document.write( "factor the right side of the equation to get 0 = (b-8) * (b-2)\r<BR>\n" );
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document.write( "solve for b to get b = 8 or b = 2.\r<BR>\n" );
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document.write( "go back to the first equation of (a+b)/2 = 5.\r<BR>\n" );
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document.write( "multiply both sides of this eqution to get a+b = 10.\r<BR>\n" );
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document.write( "when b = 8, solve for a to get a = 2.\r<BR>\n" );
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document.write( "when b = 2, solve for a to get a = 8.\r<BR>\n" );
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document.write( "that should do it.\r<BR>\n" );
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document.write( "you either have a = 2 and b = 8, or you have a = 8 and b = 2.\r<BR>\n" );
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document.write( "let's first work with a = 2 and b = 8.\r<BR>\n" );
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document.write( "the ratio of (a+b)/2 divided by sqrt(a*b) is equal to 5/4.\r<BR>\n" );
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document.write( "the ratio of b/a is 8/2 which simplifies to 4/1.\r<BR>\n" );
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document.write( "when a = 8 and b = 2, then the ratio of (a+b)/2 divided by sqrt(a*b) is still equal to 5/4.\r<BR>\n" );
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document.write( "the ratio of a/b is 8/2 which simplifies to 4/1.\r<BR>\n" );
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document.write( "looks like your first statement is true.\r<BR>\n" );
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document.write( "the ratio of the larger nmber to the smaller number is 4 to 1 in both cases.\r<BR>\n" );
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document.write( "your first statement is true.\r<BR>\n" );
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document.write( "now to your second statement.\r<BR>\n" );
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document.write( "Q.2 1. THE NUMBER OF PERMUTATION OF N OBJECTS TAEKN TOGETHER OF WHICH A OBJECTS ARE SAME TYPE AND B OBJECTS ARE OF SAME TYPE AND REMAINING ARE OF SAME TYPE IS n! / a!b!(n-a-b)!\r<BR>\n" );
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document.write( "you have a object that are the same, and you have b objects that are the same and you have c objects that are the same, where c is equal to n - a - b.\r<BR>\n" );
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document.write( "the number of permutations will be n! / (a! * b! * c!).\r<BR>\n" );
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document.write( "this is the same as n! / (a! * b! * (n-a-b)!).\r<BR>\n" );
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document.write( "the statement appears to be true.\r<BR>\n" );
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document.write( "we can test it out using a simple example.\r<BR>\n" );
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document.write( "suppose you have 4 objects.\r<BR>\n" );
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document.write( "let's call the objects a,b,c,d.\r<BR>\n" );
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document.write( "since they're all different, the number of permutations of that will be 4! = 24.\r<BR>\n" );
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document.write( "the permutations are:\r<BR>\n" );
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document.write( "abcd<BR>\n" );
document.write( "abdc<BR>\n" );
document.write( "acbd<BR>\n" );
document.write( "acdb<BR>\n" );
document.write( "adbc<BR>\n" );
document.write( "adcb \r<BR>\n" );
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document.write( "bacd<BR>\n" );
document.write( "badc<BR>\n" );
document.write( "bcad<BR>\n" );
document.write( "bcda<BR>\n" );
document.write( "bdac<BR>\n" );
document.write( "bdca\r<BR>\n" );
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document.write( "cabd<BR>\n" );
document.write( "cadb<BR>\n" );
document.write( "cbad<BR>\n" );
document.write( "cbda<BR>\n" );
document.write( "cdab<BR>\n" );
document.write( "cdba\r<BR>\n" );
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document.write( "dabc<BR>\n" );
document.write( "dacb<BR>\n" );
document.write( "dbac<BR>\n" );
document.write( "dbca<BR>\n" );
document.write( "dcab<BR>\n" );
document.write( "dcba\r<BR>\n" );
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document.write( "now we'll translate b into a, and d into c, so we have a,a,c,c.\r<BR>\n" );
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document.write( "the number of permutations will be reduced because there will be duplicates that have to be eliminated.\r<BR>\n" );
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document.write( "the new set of the same 4 objects that contains the duplicates will be:\r<BR>\n" );
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document.write( "aacc<BR>\n" );
document.write( "aacc<BR>\n" );
document.write( "acac<BR>\n" );
document.write( "acca<BR>\n" );
document.write( "acac<BR>\n" );
document.write( "acca \r<BR>\n" );
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document.write( "aacc<BR>\n" );
document.write( "aacc<BR>\n" );
document.write( "acac<BR>\n" );
document.write( "acca<BR>\n" );
document.write( "acac<BR>\n" );
document.write( "acca\r<BR>\n" );
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document.write( "caac<BR>\n" );
document.write( "caca<BR>\n" );
document.write( "caac<BR>\n" );
document.write( "caca<BR>\n" );
document.write( "ccaa<BR>\n" );
document.write( "ccaa\r<BR>\n" );
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document.write( "caac<BR>\n" );
document.write( "caca<BR>\n" );
document.write( "caac<BR>\n" );
document.write( "caca<BR>\n" );
document.write( "ccaa<BR>\n" );
document.write( "ccaa\r<BR>\n" );
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document.write( "this set will now be sorted so that all duplicates will be shown next to each other.\r<BR>\n" );
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document.write( "aacc	<BR>\n" );
document.write( "aacc	duplicate<BR>\n" );
document.write( "aacc	duplicate<BR>\n" );
document.write( "aacc	duplicate<BR>\n" );
document.write( "acac	<BR>\n" );
document.write( "acac	duplicate<BR>\n" );
document.write( "acac	duplicate<BR>\n" );
document.write( "acac	duplicate<BR>\n" );
document.write( "acca	<BR>\n" );
document.write( "acca	duplicate<BR>\n" );
document.write( "acca	duplicate<BR>\n" );
document.write( "acca	duplicate<BR>\n" );
document.write( "caac	<BR>\n" );
document.write( "caac	duplicate<BR>\n" );
document.write( "caac	duplicate<BR>\n" );
document.write( "caac	duplicate<BR>\n" );
document.write( "caca	<BR>\n" );
document.write( "caca	duplicate<BR>\n" );
document.write( "caca	duplicate<BR>\n" );
document.write( "caca	duplicate<BR>\n" );
document.write( "ccaa	<BR>\n" );
document.write( "ccaa	duplicate<BR>\n" );
document.write( "ccaa	duplicate<BR>\n" );
document.write( "ccaa	duplicate\r<BR>\n" );
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document.write( "all the duplicates will now be removed and all that will be left are the permutations that are unique.\r<BR>\n" );
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document.write( "aacc	<BR>\n" );
document.write( "acac	<BR>\n" );
document.write( "acca	<BR>\n" );
document.write( "caac	<BR>\n" );
document.write( "caca	<BR>\n" );
document.write( "ccaa	\r<BR>\n" );
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document.write( "the set of 24 has been reduced to 6.\r<BR>\n" );
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document.write( "this is equal to 4! / (2! * 2!) which is equal to 24/4 which is equal to 6.\r<BR>\n" );
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document.write( "the same applies to your larger problem.\r<BR>\n" );
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document.write( "you've got n objects.\r<BR>\n" );
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document.write( "you've got a that are the same and b that are the same and c that are the same.\r<BR>\n" );
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document.write( "the number of permutations will be n! / (a! * b! * c!).\r<BR>\n" );
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document.write( "if c happens to be (n - a - b), then the formula becomes n! / (a! * b! * (n-a-b)!)\r<BR>\n" );
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document.write( "the statement is true.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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