document.write( "Question 1026751: 1. Five cards are chosen from a random a standard deck of playing cards. In how many ways can the cards be chosen under each of the following condition.
\n" ); document.write( "a.) all are hearts. b.) Exactly three kings\r
\n" ); document.write( "\n" ); document.write( "2. Use mathematical induction to prove 2+2+8+..+2^n= 2(2^n -1) \r
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Algebra.Com's Answer #642050 by Edwin McCravy(20056) $\"\"$ $\"About$

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1. Five cards are chosen from a random a standard deck of
\n" ); document.write( "playing cards. In how many ways can the cards be chosen
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document.write( "13 hearts, CHOOSE 5\r\n" );
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document.write( "Answer:  13C5 =  = 1287\r\n" );
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document.write( "[5 factors in the numerator, coming down from 15, 5 factors in the\r\n" );
document.write( "denominator, coming up from 1.]  \r\n" );
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\n" ); document.write( "b.) Exactly three kings
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document.write( "Here we have two things to choose, 3 kings and 2 non-kings.\r\n" );
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document.write( "First we find the number of ways to choose 3 kings.\r\n" );
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document.write( "4 kings, choose 3.  4C3 =  = 4 \r\n" );
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document.write( "Next we find the number of ways to choose 2 non-kings.\r\n" );
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document.write( "48 non-kings, choose 2 = 48C2 =  = 1128\r\n" );
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document.write( "Now for each of the 4 ways to choose the 3 kings, there\r\n" );
document.write( "are 1128 ways to choose the 2 non-kings, so we multiply\r\n" );
document.write( "those numbers:\r\n" );
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document.write( "4*1128 = 4512\r\n" );
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\n" ); document.write( "Use mathematical induction to prove 2+2+8+..+2^n= 2(2^n -1)
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document.write( "We are to prove P(n) which is\r\n" );
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document.write( "First let's see what P(k+1) would be: [That's always the first \r\n" );
document.write( "thing to do.  Before you start an induction proof,  you should \r\n" );
document.write( "calculate P(k+1) to see where you're headed]: \r\n" );
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document.write( "To do that, replace n by k+1 in  to see what P(k+1) \r\n" );
document.write( "is, for that is what we are going for, and if we have that \r\n" );
document.write( "beforehand, we'll know when we have arrived and the proof is \r\n" );
document.write( "finished. \r\n" );
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document.write( "Substituting k+1 for n in , we have \r\n" );
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document.write( " = \r\n" );
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document.write( "Now that we know what P(k+1) is, we know where we're going, and we'll \r\n" );
document.write( "know we have arrived if and when we get .  So now we \r\n" );
document.write( "can start the proof: \r\n" );
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document.write( "P(1):  substitute n=1,  = 2, which is true, because the \r\n" );
document.write( "first term is 2. \r\n" );
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document.write( "Assume P(k):   \r\n" );
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document.write( "We add the next term  to both sides:\r\n" );
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document.write( "and now we see that we get the same P(k+1) as the one that we found \r\n" );
document.write( "in the beginning that we were going for.  So the proof is finished. \r\n" );
document.write( "So since P(1) is true, P(1) proves P(2), P(2) proves P(3), P(3)\r\n" );
document.write( "proves P(4), etc., etc., ad infinitum. \r\n" );
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document.write( "Edwin

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