Hello sorry i am a student who sucks at math i do have a father but can't be found nowhere for a simple question to ask i have no one but happy to found this site maybe someone out there could help me explain in a more detail way how this combinatorics works

How many arrangements can be made out of the letters of the word YIELD with the vowels not being separated

possible solution are:

20 24 48 52 58

can anyone explain then to me how it is solve so i can solve the other rest problem i am good at learning it's just that i just need to see a sample.

Consider the four blocks $Y$, $IE$, $L$ and $D$. These can be arranged in $4! = 24$ ways. Moreover the vowels in the vowels block can be arranged in $2! = 2$ ways. In total there are therefore $2 \cdot 24 = 48$ ways to arrange the letters.
Sometimes we consider $Y$ to be a vowel, in this case the blocks are $YIE$, $L$ and $D$, and a similar calculation yields an answer of $36$, so the problem setter probably did not consider $Y$ as a vowel.