document.write( "Question 1192276: a standard deck of cards has 52 cards composed of 13 hearts, 13 diamonds, 13 spades, and 13 clubs. in how many ways can hands composed of 5 cards be drawn if they must contain exactly three diamonds? \n" ); document.write( "
Algebra.Com's Answer #824187 by math_tutor2020(3817)\"\" \"About 
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\n" ); document.write( "n = 13 diamonds
\n" ); document.write( "r = 3 diamonds to select\r
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\n" ); document.write( "\n" ); document.write( "Order doesn't matter with card hands.
\n" ); document.write( "Use the nCr combination formula.
\n" ); document.write( "n C r = (n!)/(r!(n-r)!)
\n" ); document.write( "13 C 3 = (13!)/(3!*(13-3)!)
\n" ); document.write( "13 C 3 = (13!)/(3!*10!)
\n" ); document.write( "13 C 3 = (13*12*11*10!)/(3!*10!)
\n" ); document.write( "13 C 3 = (13*12*11)/(3!)
\n" ); document.write( "13 C 3 = (13*12*11)/(3*2*1)
\n" ); document.write( "13 C 3 = (1716)/(6)
\n" ); document.write( "13 C 3 = 286
\n" ); document.write( "There are 286 ways to pick the three diamond cards in any order.\r
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\n" ); document.write( "\n" ); document.write( "The remaining n = 52-13 = 39 non-diamond cards are then picked from to select r = 2 more cards.
\n" ); document.write( "Plug n = 39 and r = 2 into the nCr formula
\n" ); document.write( "n C r = (n!)/(r!(n-r)!)
\n" ); document.write( "39 C 2 = (39!)/(2!*(39-2)!)
\n" ); document.write( "39 C 2 = (39!)/(2!*37!)
\n" ); document.write( "39 C 2 = (39*38*37!)/(2!*37!)
\n" ); document.write( "39 C 2 = (39*38)/(2!)
\n" ); document.write( "39 C 2 = (39*38)/(2*1)
\n" ); document.write( "39 C 2 = (1482)/(2)
\n" ); document.write( "39 C 2 = 741
\n" ); document.write( "There are 741 ways to pick the other two non-diamond cards.\r
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\n" ); document.write( "\n" ); document.write( "We have 286*741 = 211,926 ways to form a five card hand of exactly three diamonds.
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