So let’s say we want to find how many ways we can put 20 balls into 7 boxes, where the boxes and balls are indistinguishable. For this, we can just use stars and bars, which would be 26choose6. But what would be the answer if the boxes are different? Should I multiply by 6! or 7 factorial? Thanks!

Compute (Express your answer as a common fraction.)

Edit 1: Multiplication by for cleaner answer

yo guys i am stuck so a factorial is n! = n(n-1)...(1) how do i know that it counts all cases for the possible rearrangements for n elements given all distinct?

is there a way to find the sum of factorials without calculating each one
like is there a way to compute 1! + 2! + 3! + 4! + ... + n!

Theoretically, ∞ factorial is ∞ × (∞ – 1)! = ∞ × ∞! = ∞ × ∞ × (∞ – 1)! = ∞^2 × ∞!

Theoretically, you can repeat this process an infinite amount of times meaning that ∞!=∞^∞

What do you guys think of this logic?

n! = n × (n – 1) × (n – 2) × … × 1

That is the formula for factorials. I heard that 0 factorial is 1. However if we plug 0 in for n we get 0 times something, which must be 0. So how is 0! 1? I think 0 is the most confusing number because it breaks math in so many ways. Maybe ancient mathematicians were on to something when they didn’t think 0 was an actual number. Lol but seriously why on earth is 0! equal to 1?

Any operation is allowed. I want to see if it is possible.
Rules are though:
- The only number you can have is four. Only 1 four is allowed.
- You can use any operation you'd like. (you can use mods, factorials, floors, anything!)

Compute the first few digits of

So I've been pondering about the actual amount of , and some people think its , while some people say its Can someone tell me why their answer is or ?

How would someone write factorials and using product notation?