How many possible distinct arrangements are there of the letters in the word DOG?
We could list them: DOG, DGO, ODG, OGD, GDO, GOD.  Or we could have noticed that we have 3 ways to pick the first letter, 2 ways left to pick the second, and 1 way to pick the third, for a total of $3\cdot2\cdot 1 = \boxed{6}$ ways.
6
Level 1
Prealgebra
Problem: How many possible distinct arrangements are there of the letters in the word DOG?
Solution:<|START_OF_LATENT|><|PRIOR_PREFIX|>To determine the number of distinct arrangements of the letters in the word DOG, we first recognize that the word consists of three unique letters: D, O, and G. The total number of arrangements of n distinct objects is given by the factorial of n, denoted as n!. In this case, n equals 3 because there are three letters.

We calculate the factorial of 3 as follows:
3! = 3 × 2 × 1 = 6.

This calculation shows that there are 6 different ways to arrange the letters D, O, and G. Each arrangement is distinct because all letters are unique. Therefore, the final answer is 6.<|END_OF_LATENT|> The final answer is $6$.