This is covered under the subject of "Permutations" ... a Permutation is like an ordinary combination but where the *order* matters

If I have understood your question correctly, you are asking how many ways can you arrange (without repeats) 4 letters chosen from the 7 letters "ATERING" (because you are told you have to choose the "W" and place it at the end)

So let us ignore the "W", then, and think about how many permutations of 4 you could have in 7

Now, think about this: how many combinations of 1 could be found in 7? Just 7 (A,T,E,R,I,N,G)

and how how many combinations of 2 could be found? 7×6 = 42 (because after having chosen one letter there are only 6 left to choose from)

and how how many combinations of 3 could be found? 7×6×5 = 210

and how how many combinations of 4 could be found? 7×6×5×4 = 840

And that is your answer

BTW There is a formula for the total number of permutations: P(n, r) = n! / (n-r)! , where "!" means factorial (which is calculated like this 4!=4×3×2×1)

In your case n=7 and r=4, so P = 7!/(7-4)! = 7!/3! = (7×6×5×4×3×2×1) / (3×2×1) = 7×6×5×4 = 840

And of course, you could always just list all possibilities. Let me see: ATERW, ATREW, ARTEW, RETAW ... *fades into distance*