Q.

The problem asks for the number of ways to arrange the letters of the word "NALGONDA" while ensuring that the letters G, O, and D appear in that specific order (G before O and O before D).

Step-by-Step Explanation:

The word "NALGONDA" contains 8 letters in total: N, A, L, G, O, N, D, A.

First, we identify the repetitions in the letters:

• The letter "N" appears twice.
• The letter "A" appears twice.
• The other letters (L, G, O, D) appear once each.

We are asked to arrange the letters such that the letters G, O, and D always appear in that order. So, while arranging the word, we treat the letters G, O, and D as if they form a block. The key is that these three letters must always appear in the order G → O → D, but we can have other letters (N, A, L, N, A) filling the remaining spots.

Step 1: Total Number of Positions

Since we have 8 positions in total for the 8 letters in "NALGONDA," and G, O, D must appear in a specific order, we need to find how many ways we can choose 3 positions for G, O, and D.

The number of ways to select 3 positions from 8 available positions is given by the combination formula:

        C(n, r) = n! / (r! * (n - r)!)

Here, n = 8 (total positions), and r = 3 (the positions for G, O, and D). Therefore, the number of ways to select the positions for G, O, and D is:

        C(8, 3) = 8! / (3! * (8 - 3)!) = 8! / (3! * 5!) = (8 × 7 × 6) / (3 × 2 × 1) = 56

Step 2: Arranging the Remaining Letters

Now that we have chosen positions for G, O, and D, we have 5 remaining positions that must be filled with the letters N, A, L, N, A.

The letters N and A are repeated, so the number of ways to arrange these 5 remaining letters, accounting for the repetition of N and A, is given by the formula for permutations of multiset:

        P = 5! / (2! * 2!)

Here, 5! is the total number of ways to arrange 5 letters, and 2! * 2! accounts for the repetitions of N and A. Thus, we calculate:

        P = 5! / (2! * 2!) = (5 × 4 × 3 × 2 × 1) / (2 × 1 × 2 × 1) = 120 / 4 = 30

Step 3: Final Calculation

To find the total number of ways to arrange the letters of "NALGONDA" such that G, O, and D are in that order, we multiply the number of ways to choose the positions for G, O, and D (56) by the number of ways to arrange the remaining letters (30):

        Total Ways = 56 × 30 = 1680
    

Conclusion:

Therefore, the total number of ways to arrange the letters of "NALGONDA" such that the letters G, O, and D appear in that specific order is 1680.