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Permutations And Combinations

Question

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If (n+1)! = 30 (n - 1)!, find value of n.

Solution

$space space left parenthesis straight n plus 1 right parenthesis factorial space equals space 30 space left parenthesis straight n minus 1 right parenthesis factorial$
$rightwards double arrow$ $left parenthesis straight n plus 1 right parenthesis space left parenthesis straight n right parenthesis space left square bracket left parenthesis straight n minus 1 right parenthesis factorial right square bracket space equals space 30 space left parenthesis straight n minus 1 right parenthesis factorial$

rightwards double arrow

(n + 1) (n) = 30

rightwards double arrow

straight n squared plus straight n minus 30 space equals space 0

rightwards double arrow

(n + 6) (n - 5) = 0

rightwards double arrow

space space space space space space space space space space straight n equals negative 6 space not an element of space straight N comma space straight n space equals space 5 space element of space straight N

Hence, n = 5

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