document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1087064>Question 1087064</A>: <I><SPAN STYLE=\"word-break: break-all;\"> find the probability of getting vowels in the first, third, and sixth place when all the letters of the word ORANGE are arranged in all possible ways. </SPAN>\n" );
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document.write( "Find the probability of getting vowels in the first, third, and sixth place when all the letters of the word ORANGE are arranged in all possible ways.<BR>\n" );
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document.write( "Solution:<BR>\n" );
document.write( "Examine problem: word ORANGE has 3 distinct vowels (OAE) and 3 distinct consonants (RNG).<BR>\n" );
document.write( "To find the number of arrangements with three vowels in specific places, we first place the vowels, one by one, in 3*2*1=6 ways.  Similarly there are 6 ways to place the consonants.<BR>\n" );
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document.write( "* The 6 ways are obtained as follows:<BR>\n" );
document.write( "  for the first position, we have three choices of letters<BR>\n" );
document.write( "  for the second position, we have only two letters left, since one was used <BR>\n" );
document.write( "       earlier for the first position.<BR>\n" );
document.write( "  similarly, for the sixth position, we have already used up 2 of three <BR>\n" );
document.write( "       letters, so there is only one choice left.  <BR>\n" );
document.write( "  Therefore, the total number of choices to fill the vowels is 3*2*1=6, or<BR>\n" );
document.write( "       number of arrangements = 3!<BR>\n" );
document.write( "  The same principle can be applied to filling the consonants.<BR>\n" );
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document.write( "There is a total of 6*6=36 ways to place vowels and consonants in designated locations.<BR>\n" );
document.write( "The total number of \"words\" formed using the six distinct letters is 6*5*4*3*2*1=720<BR>\n" );
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document.write( "Thus the probability of getting vowels in specific places from a random word is 36/720=1/20<BR>\n" );
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document.write( " </SPAN>\n" );
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