document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1202343>Question 1202343</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Brenda has a jar that contains 5 blue tacks and 7 red tacks. She draws 3 tacks at random without replacement. What is the probability that all 3 are red? </SPAN>\n" );
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document.write( "Answer:  <font color=red>7/44</font>\r<BR>\n" );
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document.write( "Explanation:\r<BR>\n" );
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document.write( "I'll replace \"tacks\" with \"marbles\" since they are more friendly to fingers that reach in blindly to select at random. <BR>\n" );
document.write( "Also, marbles are typically used in problems such as these.\r<BR>\n" );
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document.write( "Let's count the number of ways to select 3 red marbles.\r<BR>\n" );
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document.write( "We have n = 7 red marbles and r = 3 selections.<BR>\n" );
document.write( "Use the nCr combination formula.<BR>\n" );
document.write( "This formula is used because order doesn't matter.<BR>\n" );
document.write( "n C r = (n!)/(r!(n-r)!)<BR>\n" );
document.write( "7 C 3 = (7!)/(3!*(7-3)!)<BR>\n" );
document.write( "7 C 3 = (7!)/(3!*4!)<BR>\n" );
document.write( "7 C 3 = (7*6*5*4!)/(3!*4!)<BR>\n" );
document.write( "7 C 3 = (7*6*5)/(3!)<BR>\n" );
document.write( "7 C 3 = (7*6*5)/(3*2*1)<BR>\n" );
document.write( "7 C 3 = 210/6<BR>\n" );
document.write( "7 C 3 = 35\r<BR>\n" );
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document.write( "The value 35 can be found in Pascal's Triangle.<BR>\n" );
document.write( "Look at the row that starts with 1,7,...<BR>\n" );
document.write( "Count 4 spaces to the right to arrive at 35. <BR>\n" );
document.write( "We count four spaces (instead of three) because the start index is r = 0.\r<BR>\n" );
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document.write( "There are 35 ways to select all three red marbles, from a candidate pool of seven red overall.\r<BR>\n" );
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document.write( "We now need to find how many ways there are to select any three marbles red and/or blue.<BR>\n" );
document.write( "n = 5 blue + 7 red = 12 total<BR>\n" );
document.write( "r = 3 selections<BR>\n" );
document.write( "n C r = (n!)/(r!(n-r)!)<BR>\n" );
document.write( "12 C 3 = (12!)/(3!*(12-3)!)<BR>\n" );
document.write( "12 C 3 = (12!)/(3!*9!)<BR>\n" );
document.write( "12 C 3 = (12*11*10*9!)/(3!*9!)<BR>\n" );
document.write( "12 C 3 = (12*11*10)/(3!)<BR>\n" );
document.write( "12 C 3 = (12*11*10)/(3*2*1)<BR>\n" );
document.write( "12 C 3 = 1320/6<BR>\n" );
document.write( "12 C 3 = 220<BR>\n" );
document.write( "This value can also be found in Pascal's Triangle.\r<BR>\n" );
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document.write( "We found there are...<ul><li>35 ways to get 3 red marbles.</li><li>220 ways to select any 3 marbles (red and/or blue).</li></ul>Divide those values<BR>\n" );
document.write( "35/220 = (5*7)/(5*44) = <font color=red>7/44</font>\r<BR>\n" );
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document.write( "7/44 = 0.1590909 = 15.90909% approximately<BR>\n" );
document.write( "The \"90\" portion repeats forever.\r<BR>\n" );
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document.write( "Therefore, Brenda has about a 16% chance of getting all red marbles.<BR>\n" );
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