Permutations And Combinations

Question
CBSEENMA11014102

Find r, if 

space space space space 5. space straight P presuperscript 4 subscript straight r space equals space 6. space straight P presuperscript 5 subscript straight r minus 1 end subscript

Solution

                                  5 times space straight P presuperscript 4 subscript straight r space equals space 6 times space straight P presuperscript 5 subscript straight r minus 1 end subscript
rightwards double arrow                 5 times space fraction numerator 4 factorial over denominator left parenthesis 4 minus straight r right parenthesis factorial end fraction space equals space 6 times space fraction numerator 5 factorial over denominator left parenthesis 5 minus straight r plus 1 right parenthesis factorial end fraction
rightwards double arrow                  5 times fraction numerator 4 factorial over denominator left parenthesis 4 minus straight r right parenthesis factorial end fraction space equals space 6 times space fraction numerator 5 factorial over denominator left parenthesis 6 minus straight r right parenthesis factorial end fraction
rightwards double arrow                 5 times fraction numerator 4 factorial over denominator left parenthesis 4 minus straight r right parenthesis factorial end fraction space equals space fraction numerator 6 cross times 5 cross times 4 factorial over denominator left parenthesis 6 minus straight r right parenthesis space left parenthesis 5 minus straight r right parenthesis space left parenthesis 4 minus straight r right parenthesis factorial end fraction
rightwards double arrow                                 5 equals fraction numerator 30 over denominator left parenthesis 6 minus straight r right parenthesis space left parenthesis 5 minus straight r right parenthesis end fraction
rightwards double arrow                                  1 equals fraction numerator 6 over denominator left parenthesis 6 minus straight r right parenthesis space left parenthesis 5 minus straight r right parenthesis end fraction
rightwards double arrow                straight r squared minus 11 straight r plus 30 space equals space 6
rightwards double arrow                 straight r squared minus 11 straight r plus 24 equals 0
rightwards double arrow                  (r - 8) (r - 3) = 0
rightwards double arrow                     r = 8,  r = 3
But,  r(=8) > n (= 4)                    [Not valid]
Hence, r = 3

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