To solve the problem of how many permutations of the letters in the word "MADHUDANI" do not begin with M but end with I, we can follow these steps:
### Step 1: Identify the letters and their frequencies
The word "MADHUDANI" consists of 9 letters: M, A, D, H, U, D, A, N, I.
- The letter A appears 2 times.
- The letter D appears 2 times.
- The letters M, H, U, N, and I each appear 1 time.
### Step 2: Fix the last letter as I
Since we want the permutations to end with I, we can fix I at the end. This leaves us with the letters M, A, D, H, U, D, A, N to arrange.
### Step 3: Count the remaining letters
After fixing I, we have 8 letters left: M, A, D, H, U, D, A, N.
### Step 4: Calculate the total permutations of the remaining letters
The total number of permutations of these 8 letters can be calculated using the formula for permutations of multiset:
\[
\text{Permutations} = \frac{n!}{p_1! \times p_2! \times \ldots \times p_k!}
\]
where \( n \) is the total number of letters, and \( p_1, p_2, \ldots, p_k \) are the frequencies of the repeated letters.
Here, we have:
- Total letters (n) = 8
- A appears 2 times
- D appears 2 times
Thus, the number of permutations is:
\[
\text{Permutations} = \frac{8!}{2! \times 2!}
\]
### Step 5: Calculate \( 8! \) and \( 2! \)
Calculating \( 8! \):
\[
8! = 40320
\]
Calculating \( 2! \):
\[
2! = 2
\]
### Step 6: Substitute into the formula
Now substituting these values into the formula:
\[
\text{Permutations} = \frac{40320}{2 \times 2} = \frac{40320}{4} = 10080
\]
### Step 7: Exclude permutations that start with M
Next, we need to exclude the permutations that start with M. If M is fixed at the start, the remaining letters to arrange are A, D, H, U, D, A, N, I (with I still fixed at the end).
Now we have 7 letters left: A, D, H, U, D, A, N.
The number of permutations of these 7 letters is:
\[
\text{Permutations} = \frac{7!}{2! \times 2!}
\]
Calculating \( 7! \):
\[
7! = 5040
\]
Thus, the number of permutations starting with M is:
\[
\text{Permutations} = \frac{5040}{2 \times 2} = \frac{5040}{4} = 1260
\]
### Step 8: Final calculation
Finally, we subtract the permutations that start with M from the total permutations that end with I:
\[
\text{Required Permutations} = 10080 - 1260 = 8820
\]
### Conclusion
The total number of permutations of the letters of the word "MADHUDANI" that do not begin with M but end with I is **8820**.
