Determine the number of ways to arrange the letters of the word TARTAR.
First we count the arrangements if all the letters are unique, which is $6!$. Then since the T's, A's, and the R's are not unique, we divide by $2!$ thrice for the arrangements of T's, A's, and R's, for an answer of $\dfrac{6!}{2! \times 2! \times 2!} = \dfrac{720}{8} = \boxed{90}$.
90
Level 3
Counting & Probability
Problem: Determine the number of ways to arrange the letters of the word TARTAR.
Solution:<|START_OF_LATENT|><|PRIOR_PREFIX|>To find the number of arrangements of the letters in the word TARTAR, we first identify the total number of letters and their frequencies. The word TARTAR consists of 6 letters: T, A, R, T, A, R. 

Next, we count the occurrences of each letter: 
- T appears 2 times,
- A appears 2 times,
- R appears 2 times.

The formula for calculating the number of distinct arrangements of letters when there are repeated elements is given by the multinomial coefficient:

N = n! / (n1! * n2! * n3!)

where n is the total number of letters, and n1, n2, n3 are the frequencies of the distinct letters.

In this case, n = 6 (total letters), n1 = 2 (for T), n2 = 2 (for A), and n3 = 2 (for R). 

We calculate the factorials:
- 6! = 720,
- 2! = 2 (for T),
- 2! = 2 (for A),
- 2! = 2 (for R).

Now we substitute these values into the formula:

N = 6! / (2! * 2! * 2!) = 720 / (2 * 2 * 2) = 720 / 8 = 90.

Thus, the total number of distinct arrangements of the letters in the word TARTAR is 90.<|END_OF_LATENT|> The final answer is $90$.