document.write( "Question 1201482: This exercise refers to a standard deck of playing cards. Assume that 5 cards are randomly chosen from the deck.
\n" ); document.write( "How many hands contain 2 kings?\r
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Algebra.Com's Answer #835863 by math_tutor2020(3817)\"\" \"About 
You can put this solution on YOUR website!

\n" ); document.write( "Answer: 103,776
\n" ); document.write( "This number is slightly smaller than 104 thousand.\r
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\n" ); document.write( "\n" ); document.write( "Explanation:\r
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\n" ); document.write( "\n" ); document.write( "We can use the nCr combination formula
  • x = 4C2 = 6 ways to select the two kings in any order (see the next section below for the steps)
  • y = 48C3 = 17296 ways to select the three other cards that aren't a king in any order (eg: Queen of diamonds, Jack of clubs, 9 of hearts)
Therefore we have x*y = 6*17296 = 103,776 different five-card hands that consist of exactly 2 kings. Order doesn't matter.\r
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\n" ); document.write( "\n" ); document.write( "Example hand:
\n" ); document.write( "King of clubs, King of diamonds, 2 of spades, 3 of hearts, 8 of clubs
\n" ); document.write( "The order does not matter.\r
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\n" ); document.write( "\n" ); document.write( "Since order doesn't matter, we use the nCr formula (not the nPr formula) to compute each part.
\n" ); document.write( "I'll show the steps for each portion.
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\n" ); document.write( "Selecting the 2 kings
\n" ); document.write( "n = 4 kings total
\n" ); document.write( "r = 2 selections
\n" ); document.write( "n C r = (n!)/(r!(n-r)!)
\n" ); document.write( "4 C 2 = (4!)/(2!*(4-2)!)
\n" ); document.write( "4 C 2 = (4!)/(2!*2!)
\n" ); document.write( "4 C 2 = (4*3*2!)/(2!*2!)
\n" ); document.write( "4 C 2 = (4*3)/(2!)
\n" ); document.write( "4 C 2 = (4*3)/(2*1)
\n" ); document.write( "4 C 2 = 12/2
\n" ); document.write( "4 C 2 = 6
\n" ); document.write( "There are 6 ways to pick the two kings in any order.
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\n" ); document.write( "Selecting the 3 other cards that aren't a king
\n" ); document.write( "n = 52-4 = 48 cards that aren't a king
\n" ); document.write( "r = 3 slots to fill
\n" ); document.write( "n C r = (n!)/(r!(n-r)!)
\n" ); document.write( "48 C 3 = (48!)/(3!*(48-3)!)
\n" ); document.write( "48 C 3 = (48!)/(3!*45!)
\n" ); document.write( "48 C 3 = (48*47*46*45!)/(3!*45!)
\n" ); document.write( "48 C 3 = (48*47*46)/(3!)
\n" ); document.write( "48 C 3 = (48*47*46)/(3*2*1)
\n" ); document.write( "48 C 3 = 103776/6
\n" ); document.write( "48 C 3 = 17296
\n" ); document.write( "There are 17296 ways to pick the three other cards that aren't a king in any order.
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\n" ); document.write( "\n" ); document.write( "Further reading about the nCr formula.
\n" ); document.write( "The third link is a calculator that specializes in this formula.
\n" ); document.write( "https://mathworld.wolfram.com/Combination.html
\n" ); document.write( "https://www.mathsisfun.com/combinatorics/combinations-permutations.html
\n" ); document.write( "https://www.calculatorsoup.com/calculators/discretemathematics/combinations.php\r
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\n" ); document.write( "\n" ); document.write( "Here is another question about card hands
\n" ); document.write( "https://www.algebra.com/algebra/homework/Permutations/Permutations.faq.question.1201443.html
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