document.write( "Question 1098954: Suppose five integers are chosen successively at random between 0 and 15, inclusive. Find the probability that:
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Algebra.Com's Answer #713591 by Edwin McCravy(20056)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "There are 16 integers from 0 to 15, inclusive. \r\n" );
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document.write( "There are three cases:\r\n" );
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document.write( "Case 1: The sequence of five choices of integers contains\r\n" );
document.write( "        no integers which are the same.\r\n" );
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document.write( "16 ways to choose the 1st integer.\r\n" );
document.write( "15 ways to choose the 2nd integer.\r\n" );
document.write( "14 ways to choose the 3rd integer.\r\n" );
document.write( "13 ways to choose the 4th integer.\r\n" );
document.write( "12 ways to choose the 5th integer.\r\n" );
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document.write( "That's (16)(15)(14)(13)(12) = 16P5 = 524160 for case 1\r\n" );
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document.write( "Case 2: exactly 1 pair of the integers are alike, and \r\n" );
document.write( "the other three integers are all different, and different\r\n" );
document.write( "from the one pair of like integers.\r\n" );
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document.write( "First we choose the 5 integers and then we order them as\r\n" );
document.write( "choice 1, choice 2,..., choice 5\r\n" );
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document.write( "We can choose the integer for the one like pair in 16 ways.\r\n" );
document.write( "We choose the other 3 integers in 15C3 = 455 ways\r\n" );
document.write( "That's (16)(455) = 7280\r\n" );
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document.write( "Now that we have chosen the set of integers to be chosen, we\r\n" );
document.write( "now find the number of ways in which each set can be arranged\r\n" );
document.write( "in a sequence. \r\n" );
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document.write( "Each of those 7280 combinations can be arranged in the \r\n" );
document.write( "same number of ways as the number of distinguishable \r\n" );
document.write( "permutations of the word THERE, which has 2 \r\n" );
document.write( "indistinguishable E's. which is \"5%21%2F2%21\" or 60 ways.\r\n" );
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document.write( "So that's (7280)(60) = 436800 ways for case 2.\r\n" );
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document.write( "Case 3: exactly 2 pairs of like integers, both pairs\r\n" );
document.write( "different from each other, and the 1 remaining single \r\n" );
document.write( "integer different from either of the two pairs of like \r\n" );
document.write( "integers.\r\n" );
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document.write( "As in case 2, we first choose the integers and then we \r\n" );
document.write( "order them.\r\n" );
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document.write( "We can choose the two integers for the two pairs in \r\n" );
document.write( "16C2 = 120 ways.  We can then choose the remaining single \r\n" );
document.write( "integer in 14 ways.  That's (120)(14) =  1680 ways to \r\n" );
document.write( "choose the integers for the two pairs and the remaining \r\n" );
document.write( "integer.\r\n" );
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document.write( "Now that we have chosen the set of integers to be chosen, we\r\n" );
document.write( "now find the number of ways in which each set can be arranged\r\n" );
document.write( "in a sequence.\r\n" );
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document.write( "Each of those 1680 combinations of integers can be arranged \r\n" );
document.write( "in the same number of ways as the number of distinguishable \r\n" );
document.write( "permutations of the word LEVEL, which has 2 indistinguishable \r\n" );
document.write( "E's and 2 indistinguishable L's. which is \r\n" );
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document.write( "\"5%21%2F%282%212%21%29\" or 30 ways.\r\n" );
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document.write( "So that's (1680)(30) =  50400 ways for case 2.\r\n" );
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document.write( "Grand total for all three cases: 524160+436800+50400 = 1011360\r\n" );
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document.write( "So 1011360 is the numerator of the desired probability.\r\n" );
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document.write( "The denominator of the desired probability is the number of\r\n" );
document.write( "possible sequences of choices:\r\n" );
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document.write( "16 ways to choose the 1st integer.\r\n" );
document.write( "16 ways to choose the 2nd integer.\r\n" );
document.write( "16 ways to choose the 3rd integer.\r\n" );
document.write( "16 ways to choose the 4th integer.\r\n" );
document.write( "16 ways to choose the 5th integer.\r\n" );
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document.write( "So the denominator of the desired probability is 165 \r\n" );
document.write( "or 1048576\r\n" );
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document.write( "So the desired probability is \"1011360%2F1048576\" which reduces\r\n" );
document.write( "to \"31605%2F32768\", rounded to 4 decimal places is 0.9645\r\n" );
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document.write( "Edwin
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