Simplify the Factorial Expression.
(3n+1)!*3!/(3n-2)!
(3n+1)!*3!/(3n-2)!
= (3n+1)(3n)(3n-1)(3n-2)!3!/(3n-2)!
= 18n(9n^1 - 1)
To simplify the given factorial expression, we can start by breaking down each factorial term and then cancel out common factors.
The given expression is (3n + 1)! * 3! / (3n - 2)!
Let's first evaluate each factorial term separately:
(3n + 1)! = (3n + 1)(3n)(3n - 1)(3n - 2)(3n - 3)(3n - 4)...(6)(5)(4)(3)(2)(1)
3! = 3 * 2 * 1 = 6
(3n - 2)! = (3n - 2)(3n - 3)(3n - 4)...(6)(5)(4)(3)(2)(1)
Now, let's cancel out the common factors:
(3n + 1)! / (3n - 2)! = (3n + 1)(3n)(3n - 1)(3n - 2)(3n - 3)(3n - 4)...(6)(5)(4)(3)(2)(1) / (3n - 2)(3n - 3)(3n - 4)...(6)(5)(4)(3)(2)(1)
On the numerator and denominator, we observe that all the factors from (3n - 2) to 1 cancel out. This leaves us with:
(3n + 1)(3n)(3n - 1) / 1
Simplifying further, we have:
(3n + 1)(3n)(3n - 1) = 9n^3 - 3n^2 - 3n + 1
Finally, the simplified expression is:
(9n^3 - 3n^2 - 3n + 1) * 6 = 54n^3 - 18n^2 - 18n + 6