Understanding the Problem
To find the arrangements of the word 'Haryana' while keeping the letters 'e' and 'd' together, we treat 'ed' as a single unit or block.
Step 1: Treating 'ed' as a Block
- The word 'Haryana' has 7 letters: H, a, r, y, a, n, a.
- To incorporate 'e' and 'd' together, we first consider 'ed' as one block.
Step 2: Counting the Total Letters
- The letters we consider now are H, a, r, y, a, n, a, and 'ed'.
- This gives us a total of 6 units: H, a, r, y, n, and 'ed'.
Step 3: Accounting for Repeated Letters
- In the arrangement, the letter 'a' appears 3 times.
- The formula for arrangements is given by the total permutations divided by the permutations of the repeated items.
Step 4: Using the Permutation Formula
- The formula to calculate the arrangements is:
Total arrangements = (Number of units)! / (Number of repeated letters)!
- Here, it will be:
Total arrangements = 6! / 3!
Step 5: Calculation
- 6! = 720
- 3! = 6
- Therefore, Total arrangements = 720 / 6 = 120.
Conclusion
The total number of arrangements of the word 'Haryana' keeping 'e' and 'd' together is 120.