Understanding the Problem
To solve the problem of arranging the word "ARRANGE" such that no two 'R's appear together, we must use the Gap Method. This technique is ideal for constraints where specific items must not be adjacent.
Step 1: Identify the Letters
The word "ARRANGE" consists of 7 letters: A, R, R, A, N, G, E.
- Total letters = 7
- Count of R's = 2
- Remaining letters = A, A, N, G, E (5 letters)
Step 2: Arrange the Non-Restricted Letters
We first arrange the letters that do not have restrictions: A, A, N, G, E. Since there are two 'A's, we use the formula for permutations with identical items: $\frac{n!}{p!}$ Arrangements = $\frac{5!}{2!} = \frac{120}{2} = 60$.
Step 3: Use the Gap Method
We now place the two 'R's in the gaps created by the 5 letters. Let the letters be $X_1, X_2, X_3, X_4, X_5$. The possible gaps are: _ $X_1$ _ $X_2$ _ $X_3$ _ $X_4$ _ $X_5$ _ Counting the slots, there are 6 possible gaps for the 2 'R's.
Step 4: Calculate Placements
We need to choose 2 gaps out of 6 for the two 'R's: Number of ways to choose and arrange = $^6P_2 = \frac{6!}{(6-2)!} = 6 \times 5 = 30$.
Step 5: Final Result
Multiply the arrangements of the fixed letters by the number of valid gap placements: Total = $60 \times 30 = 180$.
Thus, there are 180 ways to arrange "ARRANGE" such that no two 'R's are adjacent.