document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1201830>Question 1201830</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A student has to choose an elective class for the last semester of high school. Her counselor suggested 9 electives, 7 of which were technology classes.<BR>\n" );
document.write( "If the student randomly chose to read about 6 of the electives, what is the probability that all of them are technology classes? </SPAN>\n" );
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document.write( "Answer:  <font color=red size=4>1/12</font><BR>\n" );
document.write( "1/12 = 0.083333 approximately\r<BR>\n" );
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document.write( "Explanation:\r<BR>\n" );
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document.write( "Let's find out how many ways there are to pick 6 technology classes to read about.<BR>\n" );
document.write( "n = 7 technology classes total<BR>\n" );
document.write( "r = 6 slots to fill<BR>\n" );
document.write( "The order of the classes doesn't matter.<BR>\n" );
document.write( "Use the nCr combination formula.<BR>\n" );
document.write( "n C r = (n!)/(r!(n-r)!)<BR>\n" );
document.write( "7 C 6 = (7!)/(6!*(7-6)!)<BR>\n" );
document.write( "7 C 6 = (7!)/(6!*1!)<BR>\n" );
document.write( "7 C 6 = (7*6!)/(6!*1!)<BR>\n" );
document.write( "7 C 6 = (7)/(1!)<BR>\n" );
document.write( "7 C 6 = 7/1<BR>\n" );
document.write( "7 C 6 = 7<BR>\n" );
document.write( "There are 7 ways to pick 6 technology classes to read about.\r<BR>\n" );
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document.write( "That's a fairly long-winded approach.<BR>\n" );
document.write( "Is there a shortcut, and a way to avoid that formula?<BR>\n" );
document.write( "Yes there is.<BR>\n" );
document.write( "Imagine there are 2 groups: A and B<BR>\n" );
document.write( "Group A represents what you pick (those 6 items to read). <BR>\n" );
document.write( "Group B is what you don't pick: that 1 class left out.<BR>\n" );
document.write( "At this point it should be fairly clear that there are 7 ways to fill group B, which means there are 7 ways to fill group A.\r<BR>\n" );
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document.write( "Now let's determine how many ways there are to select 6 classes, where some of which may not be technology classes.<BR>\n" );
document.write( "n = 9 classes total<BR>\n" );
document.write( "r = 6 slots<BR>\n" );
document.write( "Use the nCr formula<BR>\n" );
document.write( "n C r = (n!)/(r!(n-r)!)<BR>\n" );
document.write( "9 C 6 = (9!)/(6!*(9-6)!)<BR>\n" );
document.write( "9 C 6 = (9!)/(6!*3!)<BR>\n" );
document.write( "9 C 6 = (9*8*7*6!)/(6!*3!)<BR>\n" );
document.write( "9 C 6 = (9*8*7)/(3!)<BR>\n" );
document.write( "9 C 6 = (9*8*7)/(3*2*1)<BR>\n" );
document.write( "9 C 6 = 504/6<BR>\n" );
document.write( "9 C 6 = 84<BR>\n" );
document.write( "There are 84 ways to select 6 classes from a candidate pool of 9.\r<BR>\n" );
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document.write( "Rephrased another way:<BR>\n" );
document.write( "There are 9*8*7 = 504 permutations<BR>\n" );
document.write( "We divide by 3*2*1 = 6 to adjust the fact order doesn't matter (something like ABC is the same as BAC). <BR>\n" );
document.write( "There are 6 ways to arrange any trio of items.<BR>\n" );
document.write( "Therefore, we arrive at 504/6 = 84 combinations.\r<BR>\n" );
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document.write( "Each nCr value can be found in Pascal's Triangle. <BR>\n" );
document.write( "For instance, the value 84 is in the row that starts with 1,9,...\r<BR>\n" );
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document.write( "There's a lot to take in, so read each subsection again to make sure you understand what is going on.\r<BR>\n" );
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document.write( "To recap:<ul><li>7 ways to pick the 6 technology classes </li><li>84 ways to pick any 6 classes (technology or not)</li></ul>Order does not matter.\r<BR>\n" );
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document.write( "Divide those values to get the probability we want.<BR>\n" );
document.write( "7/84 = (1*7)/(12*7) = <font color=red>1/12</font> is the probability that all 6 selections are technology classes.\r<BR>\n" );
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document.write( "Use a calculator to get its approximate value<BR>\n" );
document.write( "1/12 = 0.083333<BR>\n" );
document.write( "It might be better to stick to the fraction form since it is exact. <BR>\n" );
document.write( "Be sure to follow your teacher's instructions.<BR>\n" );
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