Permutation formula is ambiguous for this question. The number of permutations of n distinct objects taken r at a time is nPr.
You have four letters, a, b, c, d. Consider the number of permutations that are possible by taking two letters at a time from these four.
Can someone explain the disparity between these two approaches using tree diagram terminology? The answer below lists 12 possible branches, implying 12 $\times$ 1, but in the comments it is found using 4 $\times$ 3. Which is right? or more right?
 A: @jiniyt You are using 4P2 which gives us the number of permutation of 4 distinct objects ($ a, b, c$ and $d$) taken $2$ at a time(you are taking any two of the four available letters and permuting them). Due to which you get - $ab, ba, ac, ca, ad, da, bc, cb, bd, bd, cd$ and $dc$. 
In order to get - abcd, abdc, acbd, acdb, adbc, adcb, bacd, badc, bcda, bcad, bdac, bdca, cadb, cabd, cbda, cbad, cdba, cdab, dabc, dacb, dbca, dbac, dcab, dcba - the answer that you got, as you can see, you have to take all four of the four available letters and permute them. Hence, by using the formula, we get 4P4 which is nothing but $4!$ which is $24$.
If the answer is still unclear then let me know so that I can explain it to you more clearly.
EDIT -
@jiniyt I didn't have the time to make a tree diagram.
Why $3$ x $4$?
Answer - We need to make a 2 letter word. Hence we have to choose any 2 out of the four letters. So firstly, we can choose any one of the four letters which gives us 4. Second, now when we choose the second letter, one letter has already been chosen, so we can now choose only any one of the three remaining letters. That's why we have $4$ x $3$ $=$ $12.
Let me explain this to you more clearly. We have four available letters. We need to make a two letter word from these four available letters($a$, $b$, $c$, $d$). Inorder to make a two letter word from these four letters, we need to choose any two of them.
So we have ($a$, $b$, $c$, $d$) and let us choose $a$ as the first letter, that leaves us with an option to choose either ($b$, $c$, $d$) as the second letter. We can also choose $b$/$c$/$d$ as the first letters and each would leave us a different set of letters to choose from for the second letter of the two letter word.
So we can choose any one of the 4 letters as the first letter of our two letter word, so we have $4$C$1$. And we now have to choose any one of the 3 letters as our second letter of our two letter word, so we have $3$C$1$. Which gives us a total of $4$x$3$ ways to make a 2 letter word from four distinct letter. Giving us $12$ as the answer.
NOTE - I couldn't comment as I didn't have enough reputation, so I had to write this down as an answer.
