How many different arrangements can be obtained from the letters of the word
LANGUAGE so that the letter N always appears before all the A’s?

To solve the problem of how many different arrangements can be obtained from the letters of the word "LANGUAGE" such that the letter N always appears before all the A's, we can follow these steps:

1. **Count the total number of letters**: The word "LANGUAGE" has 8 letters.
2. **Identify the letters**: L, A, N, G, U, A, G, E. Notably, there are 2 A's and 2 G's.
3. **Calculate the total permutations without restrictions**: The total number of permutations of the letters in "LANGUAGE" can be calculated considering the repetitions of A and G:
   \[
   \text{Total permutations} = \frac{8!}{2! \cdot 2!} = \frac{40320}{4} = 10080
   \]

4. **Consider the restriction that N must appear before all the A's**: 
   - There are 3 positions (including N and 2 A's) that involve N and the 2 A's.
   - To ensure N appears before all the A's, we choose 1 out of the 3 positions for N. The remaining 2 positions must be for the 2 A's, which can only be arranged in 1 way (since both are A's).

   This restriction divides the total permutations by the number of ways to arrange N and 2 A's without any restriction. There are \( \frac{3!}{2!} = 3 \) unrestricted ways to arrange N and 2 A's, and only one of those arrangements satisfies N before all A's.

5. **Calculate the number of valid permutations**:
   \[
   \text{Valid permutations} = \frac{\text{Total permutations}}{3} = \frac{10080}{3} = 3360
   \]

So, the number of different arrangements of the letters in "LANGUAGE" where N appears before all the A's is \( \boxed{3360} \).

Would you like more details or have any questions? Here are some questions you might want to ask:

1. How do you calculate permutations with repetitions?
2. What if a different letter had to appear before another?
3. How do factorials work in these calculations?
4. What is the difference between permutations and combinations?
5. Can you provide examples with different words?

**Tip:** When solving permutation problems with restrictions, break down the problem step-by-step to ensure each condition is met systematically.

Learn how to calculate permutations of the word LANGUAGE where the letter N appears before all instances of A. This problem involves permutations, factorials, and combinatorial counting. Suitable for high school students studying advanced combinatorics.

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