document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1175160>Question 1175160</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A five-digit number is formed by arranging 2, 3, 4, 5, and 6 randomly without repetition. Find the probabilities of the following events<BR>\n" );
document.write( "1. The unit digit of the number is 5.<BR>\n" );
document.write( "2. T he number is an even even number. </SPAN>\n" );
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document.write( "Problem 1\r<BR>\n" );
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document.write( "We require that 5 is in the units place. So that locks that digit up. The number looks like ABCD5, where A,B,C,D are the digit placeholders for some permutation of {2,3,4,6}. We aren't multiplying the A through D values out.\r<BR>\n" );
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document.write( "For slot A, we have four choices {2,3,4,6}<BR>\n" );
document.write( "For slot B, we have three choices (one less than the number of choices in slot A)<BR>\n" );
document.write( "For slot C, we have two choices (we count down one more)<BR>\n" );
document.write( "For slot D, we have one choice\r<BR>\n" );
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document.write( "Overall, there's 4*3*2*1 = 24 different permutations here. Order matters.\r<BR>\n" );
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document.write( "If we don't fix that last digit to be a 5, then we have 5! = 5*4*3*2*1 = 120 different permutations.\r<BR>\n" );
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document.write( "The probability of getting a 5 in the units place is 24/120 = 1/5\r<BR>\n" );
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document.write( "Answer: <font color=red>1/5</font>\r<BR>\n" );
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document.write( "Problem 2\r<BR>\n" );
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document.write( "Consider slots A,B,C,D,E where E is the units digit.\r<BR>\n" );
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document.write( "We want the five digit number to be even, so that must mean slot E takes on the value of 2, 4 or 6. We have three choices here.\r<BR>\n" );
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document.write( "Then for slot D, we have 5-1 = 4 choices after we ignore whatever is selected for slot E.<BR>\n" );
document.write( "For slot C, we have 4-1 = 3 choices. We count our way down one at a time.<BR>\n" );
document.write( "For slot B, we have 3-1 = 2 choices<BR>\n" );
document.write( "For slot A, we have 2-1 = 1 choice\r<BR>\n" );
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document.write( "Multiply out those values: 3*4*3*2*1 = 3*4! = 3*24 = 72<BR>\n" );
document.write( "There are 72 ways to form a five digit even number with the values {2,3,4,5,6} and repetition is not allowed. \r<BR>\n" );
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document.write( "Divide this over 120, which was the number of ways to form a five digit number without restrictions, and we get 72/120 = 3/5\r<BR>\n" );
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document.write( "Answer: <font color=red>3/5</font><BR>\n" );
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