Question

how many arrangments of the word father are there? ​

Answer 1

$\huge \mathfrak \pink{your \: ans.}$

How many arrangements of ALIVE can be made if all of the vowels must be kept together?

Notice that ALIVE has 5 letters, 3 of which are vowels. Since the vowels must be kept together, consider them as 1 object, which, together with the L and the V gives us 3 objects to arrange.

There are 3! = 6 ways to arrange these three objects. But the three vowels themselves, although kept together, could also be arranged in 3! = 6 ways themselves.

Therefore there are 3! x 3! = 6 x 6 = 36 arrangements possible.