How many 4-letter words can be formed by slection four letters from the word UNIFORM? I don't understand how to do this at all. I know the answer is 840 but why?

Question

i have no idea how to simplify this... can anyone help?
(x+3)!/x+3

(x-2)!/x!

Answer 1

Think of filling 4 compartments with different letters from UNIFORM

__*__*__*__*

you could fill the first place with any of the 7 letters, so there are 7 ways to do that...

7*__*__*__*

now that one letter is gone, you can fill the 2nd place with any of the 6 remaining letters ....

7*6*__*__*

that leaves any of the 5 remaining letters to go into the 3rd spot

7*6*5*__*

and finally any of the remaining 4 letters can go into the last place.

7*6*5*4*

7*6*5*4 = 840

You can use this type of reasoning for most of these questions.

Answer 2

for (x+3)!/x+3

let's illustrate with a numerical example

what is 5!/5

5! means 5*4*3*2*1

and when you divide that by 5, wouldn't that cancel the leading factor of 5 leaving you with
4*3*2*1 or 4!

in the same way
(x+3)! = (x+3)(x+2)(x+1)...(2)(1)

divide that by x+3 cancels the first factor leaving
(x+2)(x+1)...(2)(1) or (x+2)!

you do the last one, let me know what you got.

Answer 3

i have no idea how to do the last one