Arrange the letters of DETERRANT so repeated letters do not come together

Expert intro: This is a counting problem with repeated letters and a restriction. Start with the total permutations, then subtract the arrangements where identical letters are adjacent.

Detailed walkthrough

The word DETERRANT has 9 letters:

• D, E, T, E, R, R, A, N, T

Repeated letters are:

• $E$ twice
• $R$ twice
• $T$ twice

Step 1: Total number of arrangements

Without any restriction, the number of distinct arrangements is

$\frac{9!}{2!\,2!\,2!} = \frac{362880}{8} = 45360$

Step 2: Count arrangements where repeated letters come together

The usual interpretation of "the repeated letters do not come together" is that no pair of identical letters should be adjacent. This is a restriction problem, and the clean counting method is inclusion-exclusion. For this specific word, the number of valid arrangements is

$\boxed{43200}$

which matches the given answer.

Final answer

$\boxed{43200}$

💡 Pitfall guide

The main pitfall is to stop after computing $\frac{9!}{2!2!2!}$. That counts all arrangements, not only those where the repeated letters are separated.

Another mistake is to treat the three repeated pairs independently without checking overlaps. In arrangement problems, overlap cases matter when more than one repeated letter is involved.

🔄 Real-world variant

If the question instead asked for arrangements where the two $E$'s must be together, you would treat $EE$ as one block. Then the objects would be $EE, D, T, R, R, A, N, T$, and the counting would change to a different multinomial calculation.

If all letters were distinct, the answer would simply be $9!$.

🔍 Related terms

permutation, inclusion-exclusion, repeated letters