document.write( "Question 1038187: The following uses the five vowels (A,E,I,O,U). \r
\n" ); document.write( "\n" ); document.write( "What is the probability that when randomly selecting a permutation of the 26 letters (assumed all uppercase) of the alphabet the following events will occur:\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "A: The first 13 letters will be in alphabetical order.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "E: The first and last letters will be B and Y.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "I: The letter I will come before both J and K.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "O: The letters O and P will be next to each other.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "U: The letters U and V will be separated by at least 23 letters.
\n" ); document.write( " \n" ); document.write( "

Algebra.Com's Answer #652922 by Edwin McCravy(20054) $\"\"$ $\"About$

You can put this solution on YOUR website!
The following uses the five vowels (A,E,I,O,U). \r
\n" ); document.write( "\n" ); document.write( "What is the probability that when randomly selecting
\n" ); document.write( "a permutation of the 26 letters (assumed all uppercase)
\n" ); document.write( "of the alphabet the following events will occur:\r
\n" ); document.write( "\n" ); document.write( "A: The first 13 letters will be in alphabetical order.
\n" ); document.write( "

\r\n" );
document.write( "It is obvious (to me) that the probability is 1/13!\r\n" );
document.write( "But if not to you, then:\r\n" );
document.write( "\r\n" );
document.write( "There are 52! arrangements of the letters of the alphabet.\r\n" );
document.write( "\r\n" );
document.write( "Consider a sample such arrangement where the first 13\r\n" );
document.write( "are in alphabetical order, say,\r\n" );
document.write( "\r\n" );
document.write( "(1). BCDGHIJPQWXYZEMLNVRKAOSUFT\r\n" );
document.write( "\r\n" );
document.write( "Among the 52!, there are 13! ways the first 13 letters \r\n" );
document.write( "can be arranged, one such being, say,\r\n" );
document.write( "\r\n" );
document.write( "(2). PIZDHBYGXCWJQEMLNVRKAOSUFT\r\n" );
document.write( "\r\n" );
document.write( "Thus (1) is the only \"successful\" case out of the 13! cases \r\n" );
document.write( "that include only the \"successful\" (1) and the other\r\n" );
document.write( "\"unsuccessful\" cases such as (2).\r\n" );
document.write( "\r\n" );
document.write( "Thus to get the number of ways the first 13 letters are in \r\n" );
document.write( "alphabetical order, we must divide 52! by 13!\r\n" );
document.write( "\r\n" );
document.write( "Thus the probability is 52!/13! ways out of 52! or\r\n" );
document.write( "\r\n" );
document.write( " =   \r\n" );
document.write( "-------------------------------\r\n" );
document.write( "

\n" ); document.write( "E: The first and last letters will be B and Y.
\n" ); document.write( "

\r\n" );
document.write( "That is 24!/26! = 1/(26*25) = 1/650,\r\n" );
document.write( "for there are 24! ways to arrange the other 24\r\n" );
document.write( "letters between B on the far left and Y on the\r\n" );
document.write( "far right.\r\n" );
document.write( "\r\n" );
document.write( "B _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ Y\r\n" );
document.write( "\r\n" );
document.write( "-------------------------------\r\n" );
document.write( "

\n" ); document.write( "I: The letter I will come before both J and K.
\n" ); document.write( "

\r\n" );
document.write( "Look at these 3! or 6 arrangements of the 26 letters:\r\n" );
document.write( "\r\n" );
document.write( "PIZDHBYGXCWJQEMLNVRKAOSUFT\r\n" );
document.write( "\r\n" );
document.write( "PIZDHBYGXCWKQEMLNVRJAOSUFT\r\n" );
document.write( "\r\n" );
document.write( "PJZDHBYGXCWIQEMLNVRKAOSUFT\r\n" );
document.write( "\r\n" );
document.write( "PKZDHBYGXCWIQEMLNVRJAOSUFT\r\n" );
document.write( "\r\n" );
document.write( "PJZDHBYGXCWKQEMLNVRIAOSUFT\r\n" );
document.write( "\r\n" );
document.write( "PKZDHBYGXCWJQEMLNVRIAOSUFT\r\n" );
document.write( "\r\n" );
document.write( "Only the first 2 out of those 6 ways does I come before\r\n" );
document.write( "both J and K.  Thus the probability is 2/6 or 1/3\r\n" );
document.write( "\r\n" );
document.write( "-----------------\r\n" );
document.write( "

\n" ); document.write( "O: The letters O and P will be next to each other.
\n" ); document.write( "

\r\n" );
document.write( "That's the 25! arrangements of these 25 things,\r\n" );
document.write( "consisting of 24 single letters and 1 letter pair: \r\n" );
document.write( "\r\n" );
document.write( "A,B,C,D,E,F,G,H,I,J,K,L,M,N,(OP),Q,R,S,T,U,V,W,X,Y,Z\r\n" );
document.write( "\r\n" );
document.write( "plus the 25! arrangements of these 25 things,\r\n" );
document.write( "also consisting of 24 single letters and 1 letter pair:\r\n" );
document.write( "\r\n" );
document.write( "A,B,C,D,E,F,G,H,I,J,K,L,M,N,(PO),Q,R,S,T,U,V,W,X,Y,Z\r\n" );
document.write( "\r\n" );
document.write( "So the probability is 2(25!/26!) = 2(1/26) = 1/13\r\n" );
document.write( "\r\n" );
document.write( "-------------------\r\n" );
document.write( "

\n" ); document.write( "U: The letters U and V will be separated by at least 23 letters.
\n" ); document.write( "

\r\n" );
document.write( "There are the 6 ways indicated below to place the U and V, \r\n" );
document.write( "and 24! ways of arranging the other 24 letters.\r\n" );
document.write( "\r\n" );
document.write( "U _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ V _ \r\n" );
document.write( "U _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ V  \r\n" );
document.write( "_ U _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ V \r\n" );
document.write( "V _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ U _ \r\n" );
document.write( "V _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ U  \r\n" );
document.write( "_ V _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ U \r\n" );
document.write( "\r\n" );
document.write( "6(24!)/(26!) = 6[1/(26*25)] = 6/(26*25) = 3(13*25) = 3/325\r\n" );
document.write( "\r\n" );
document.write( "Edwin

\n" ); document.write( "

\n" );