Find the number of permutations of the letters of the words : 'DADDY DID A DEADLY DEED'.

Answer

To find the number of permutations of the letters in the phrase "DADDY DID A DEADLY DEED", we can follow these steps:

### Step 1: Count the Total Number of Letters
First, we need to count the total number of letters in the phrase.

- The phrase "DADDY DID A DEADLY DEED" has 19 letters in total.

### Step 2: Count the Frequency of Each Letter
Next, we need to count how many times each letter appears in the phrase.

- **D** appears 5 times.
- **A** appears 3 times.
- **Y** appears 2 times.
- **I** appears 2 times.
- **L** appears 1 time.
- **E** appears 2 times.

### Step 3: Apply the Permutation Formula
The formula for permutations of a multiset (where some items are identical) is given by:

\[ \text{Number of permutations} = \frac{n!}{n_1! \cdot n_2! \cdot n_3! \cdots} \]

Where:
- \( n \) is the total number of items (letters in this case),
- \( n_1, n_2, n_3, \ldots \) are the frequencies of the identical items.

Here, we have:

- \( n = 19 \) (total letters)
- \( n_D = 5 \) (for D's)
- \( n_A = 3 \) (for A's)
- \( n_Y = 2 \) (for Y's)
- \( n_I = 2 \) (for I's)
- \( n_E = 2 \) (for E's)
- \( n_L = 1 \) (for L's)

Thus, we can write the formula as:

\[ \text{Number of permutations} = \frac{19!}{5! \cdot 3! \cdot 2! \cdot 2! \cdot 2! \cdot 1!} \]

### Step 4: Calculate the Factorials
Now we need to calculate the factorials:

- \( 19! \) is the factorial of 19.
- \( 5! = 120 \)
- \( 3! = 6 \)
- \( 2! = 2 \)
- \( 1! = 1 \)

### Step 5: Substitute and Simplify
Now we substitute these values into the formula:

\[ \text{Number of permutations} = \frac{19!}{5! \cdot 3! \cdot 2! \cdot 2! \cdot 2! \cdot 1!} = \frac{19!}{120 \cdot 6 \cdot 2 \cdot 2 \cdot 2 \cdot 1} \]

Calculating the denominator:

\[ 120 \cdot 6 = 720 \]
\[ 720 \cdot 2 = 1440 \]
\[ 1440 \cdot 2 = 2880 \]
\[ 2880 \cdot 2 = 5760 \]

So, the denominator is \( 5760 \).

### Step 6: Final Calculation
Now we can compute the final number of permutations:

\[ \text{Number of permutations} = \frac{19!}{5760} \]

This will give us the total number of distinct arrangements of the letters in the phrase "DADDY DID A DEADLY DEED".