document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1186814>Question 1186814</A>: <I><SPAN STYLE=\"word-break: break-all;\"> There are 4 sets of balls numbered 1 through 5 placed in a bowl. If 4 balls are randomly chosen without replacement, find the probability that the balls have the same number. Express your answer as a fraction in lowest terms or a decimal rounded to the nearest millionth. </SPAN>\n" );
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document.write( "There are A = 5 ways to get what we want, since there are 5 items in the set {1,2,3,4,5}. This set is repeated four times to represent the 4*5 = 20 balls.\r<BR>\n" );
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document.write( "For example, one possible desired outcome is selecting the four balls labeled \"3\". <BR>\n" );
document.write( "In other words, having this outcome: 3,3,3,3\r<BR>\n" );
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document.write( "There are n = 20 balls total and r = 4 of them are selected. Order doesn't matter, so we'll use the combination formula nCr\r<BR>\n" );
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document.write( "The notation <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large _{n} C_{r}\"> is the same as C(n,r)\r<BR>\n" );
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document.write( "C(n, r) = (n!)/( r!*(n-r)! )<BR>\n" );
document.write( "C(20, 4) = (20!)/( 4!*(20-4)! )<BR>\n" );
document.write( "C(20, 4) = (20!)/( 4!*16! )<BR>\n" );
document.write( "C(20, 4) = (20*19*18*17*16!)/( 4!*16! )<BR>\n" );
document.write( "C(20, 4) = (20*19*18*17)/( 4! ) .....  the \"16!\" terms cancel<BR>\n" );
document.write( "C(20, 4) = (20*19*18*17)/( 4*3*2*1 )<BR>\n" );
document.write( "C(20, 4) = ( 116280 )/( 24 )<BR>\n" );
document.write( "C(20, 4) = 4845<BR>\n" );
document.write( "This represents the number of ways to select the four balls.<BR>\n" );
document.write( "Let B = 4845\r<BR>\n" );
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document.write( "To summarize:<BR>\n" );
document.write( "We have A = 5 ways to get what we want (having all four balls with the same number)<BR>\n" );
document.write( "There are B = 4845 ways to select the four balls from a pool of 20\r<BR>\n" );
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document.write( "The probability of getting what we want is A/B = 5/4845 = <font color=red>1/969</font>. To reduce that fraction, I divided both parts by the GCF 5.\r<BR>\n" );
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document.write( "Use your calculator to find that 1/969 = 0.00103199174407 approximately. Rounding to the nearest millionth gets us <font color=red>0.001032</font>\r<BR>\n" );
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document.write( "Note: Rounding to the nearest millionths decimal place means the decimal answer will have 6 decimal places.  \r<BR>\n" );
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document.write( "--------------------------------------------------------------\r<BR>\n" );
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document.write( "Answer as a fraction: <font color=red>1/969</font> (exact)<BR>\n" );
document.write( "Answer as a decimal: <font color=red>0.001032</font> (approximate; rounded to 6 decimal places)<BR>\n" );
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