I know "!" means e.g. 5!=5x4x3x2x1 but how about [3(n+1)!]/[5n!]?

Can you see how n! = n(n-1)!?

or that (n+1)! = (n+1)n! ?
just like 8! = 8(7!)

then [3(n+1)!]/[5n!]
=3(n+1)n! / 5n!

To simplify the expression [3(n+1)!]/[5n!], we can use the properties of factorials.

First, let's understand the notation:
"(n+1)!" represents the factorial of (n+1), which means multiplying consecutive positive integers from 1 to (n+1).
For example, 5! = 5 x 4 x 3 x 2 x 1 = 120.

Now, let's use the property that n! = n(n-1)! to simplify (n+1)! in the expression.

(n+1)! = (n+1)n!

This property tells us that if we know the factorial of any number, we can express the factorial of the next number using multiplication. For example, if we know 8!, we can calculate 9! as 9 x 8!.

Using this property, we can simplify the expression [3(n+1)!]/[5n!] as follows:

[3(n+1)!]/[5n!] = 3(n+1)n! / 5n!

Now, we can cancel out the common factor of n! in the numerator and denominator:

= 3(n+1) / 5

Therefore, the simplified expression is 3(n+1) / 5.

It's important to note that when simplifying expressions involving factorials, it's often helpful to use the properties of factorials and simplify step by step.