Permutations and Combinations

Evaluate: when (a) n = 6, r = 2, (b) n = 6, r = 4, (c) n = 9, r = 4 - Wired Faculty

Question

Evaluate: $fraction numerator straight n factorial over denominator straight r factorial space left parenthesis straight n minus straight r right parenthesis factorial end fraction comma$ when

(a) n = 6, r = 2, (b) n = 6, r = 4, (c) n = 9, r = 4

Solution

(a) n = 6, r = 2
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(b) n = 6, r = 4
$rightwards double arrow$ $fraction numerator straight n factorial over denominator straight r factorial left parenthesis straight n minus straight r right parenthesis factorial end fraction equals fraction numerator 6 factorial over denominator 2 factorial space 4 factorial end fraction space equals space fraction numerator 6 cross times 5 cross times 4 factorial over denominator 4 factorial space 2 factorial end fraction space equals space fraction numerator 6 cross times 5 over denominator 2 cross times 1 end fraction equals 15$
(c) n = 9, r = 4
$rightwards double arrow$ $fraction numerator straight n factorial over denominator straight r factorial space left parenthesis straight n minus straight r right parenthesis factorial end fraction space equals space fraction numerator 9 factorial over denominator 4 factorial space 5 factorial end fraction equals fraction numerator 9 cross times 8 cross times 7 cross times 6 cross times 5 factorial over denominator 4 factorial space 5 factorial end fraction equals fraction numerator 9 cross times 8 cross times 7 cross times 6 over denominator 4 cross times 3 cross times 2 cross times 1 end fraction equals 126.$

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