How do you simplify $(3(n+1)! )/ (5n!)$ ?

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Answer 1 · 2015-11-09T15:35:35

${3(n+1)!}/{5n!} = {3(n+1)}/5$

The factorial of any integer $n$ is given by

$n! = n(n-1)(n-2)(n-3)...(3)(2)(1)$

Therefore, the factorial of $n+1$ would be

$(n+1)! = (n+1)(n)(n-1)(n-2)(n-3)...(3)(2)(1)$

Using these two expression in our original question,

${3(n+1)!}/{5n!} = [3(n+1)(n)(n-1)(n-2)...(3)(2)(1)]/[5(n)(n-1)(n-2)...(3)(2)(1)]$

All terms in the denominator except $5$ are cancelled out by corresponding terms in the numerator. Therefore, you finally get

${3(n+1)!}/{5n!} = {3(n+1)}/5$

How do you simplify (3(n+1)! )/ (5n!)(3(n+1)! )/ (5n!)?