Permutations and Combinations

Question

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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Permutations And Combinations

Find r, if

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

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