If the different permutations of the word PRODIGIOUS are listed as in a dictionary, then what is the rank of the word PRODIGIOUS?

Options

A. a. 200966
B. b. 609902
C. c. 12500
D. d. 24800

Answer

To find the rank of the word "PRODIGIOUS" when all its permutations are listed in dictionary order, we will follow these steps:

### Step 1: Count the letters and their frequencies
The word "PRODIGIOUS" consists of 10 letters: P, R, O, D, I, G, I, O, U, S.
- The frequency of letters is:
- P: 1
- R: 1
- O: 2
- D: 1
- I: 2
- G: 1
- U: 1
- S: 1

### Step 2: Arrange the letters in alphabetical order
The letters in alphabetical order are:
- D, G, I, I, O, O, P, R, S, U

### Step 3: Calculate permutations starting with letters before 'P'
We will calculate the permutations of the letters that come before 'P' in the alphabetical order.

1. **Starting with D**:
- Remaining letters: G, I, I, O, O, P, R, S, U (9 letters)
- Permutations = \( \frac{9!}{2! \times 2!} = \frac{362880}{4} = 90720 \)

2. **Starting with G**:
- Remaining letters: D, I, I, O, O, P, R, S, U (9 letters)
- Permutations = \( \frac{9!}{2! \times 2!} = 90720 \)

3. **Starting with I**:
- Remaining letters: D, G, I, O, O, P, R, S, U (9 letters)
- Permutations = \( \frac{9!}{2!} = \frac{362880}{2} = 181440 \)

4. **Starting with O**:
- Remaining letters: D, G, I, I, O, P, R, S, U (9 letters)
- Permutations = \( \frac{9!}{2!} = 181440 \)

### Step 4: Calculate permutations starting with 'P'
Now we will calculate the permutations starting with 'P'.

1. **Starting with PD**:
- Remaining letters: G, I, I, O, O, R, S, U (8 letters)
- Permutations = \( \frac{8!}{2! \times 2!} = \frac{40320}{4} = 10080 \)

2. **Starting with PG**:
- Remaining letters: D, I, I, O, O, R, S, U (8 letters)
- Permutations = \( \frac{8!}{2! \times 2!} = 10080 \)

3. **Starting with PI**:
- Remaining letters: D, G, I, O, O, R, S, U (8 letters)
- Permutations = \( \frac{8!}{2!} = 20160 \)

4. **Starting with PO**:
- Remaining letters: D, G, I, I, O, R, S, U (8 letters)
- Permutations = \( \frac{8!}{2!} = 20160 \)

5. **Starting with PR**:
- Remaining letters: D, G, I, I, O, O, S, U (8 letters)
- Permutations = \( \frac{8!}{2! \times 2!} = 10080 \)

6. **Starting with PS**:
- Remaining letters: D, G, I, I, O, O, R, U (8 letters)
- Permutations = \( \frac{8!}{2! \times 2!} = 10080 \)

### Step 5: Calculate permutations starting with 'PRO'
1. **Starting with PROD**:
- Remaining letters: I, I, O, O, R, S, U (7 letters)
- Permutations = \( \frac{7!}{2! \times 2!} = 2520 \)

2. **Starting with PROG**:
- Remaining letters: D, I, I, O, O, R, S, U (7 letters)
- Permutations = \( \frac{7!}{2!} = 2520 \)

3. **Starting with PROI**:
- Remaining letters: D, G, I, O, O, R, S, U (7 letters)
- Permutations = \( \frac{7!}{2!} = 2520 \)

4. **Starting with PROO**:
- Remaining letters: D, G, I, I, R, S, U (7 letters)
- Permutations = \( \frac{7!}{2!} = 2520 \)

5. **Starting with PROR**:
- Remaining letters: D, G, I, I, O, S, U (7 letters)
- Permutations = \( \frac{7!}{2! \times 2!} = 2520 \)

6. **Starting with PRS**:
- Remaining letters: D, G, I, I, O, O, U (7 letters)
- Permutations = \( \frac{7!}{2! \times 2!} = 2520 \)

### Step 6: Calculate permutations starting with 'PROD'
1. **Starting with PRODIG**:
- Remaining letters: I, O, O, R, S, U (6 letters)
- Permutations = \( \frac{6!}{2!} = 360 \)

2. **Starting with PRODIO**:
- Remaining letters: I, O, R, S, U (6 letters)
- Permutations = \( 6! = 720 \)

3. **Starting with PRODIGI**:
- Remaining letters: O, O, R, S, U (6 letters)
- Permutations = \( \frac{6!}{2!} = 360 \)

### Step 7: Calculate the rank of "PRODIGIOUS"
Now, we will sum all the permutations calculated above to find the rank of "PRODIGIOUS".

- Total permutations before "PRODIGIOUS":
- Starting with D: 90720
- Starting with G: 90720
- Starting with I: 181440
- Starting with O: 181440
- Starting with PD: 10080
- Starting with PG: 10080
- Starting with PI: 20160
- Starting with PO: 20160
- Starting with PR: 10080
- Starting with PS: 10080
- Starting with PROD: 2520
- Starting with PROG: 2520
- Starting with PROI: 2520
- Starting with PROO: 2520
- Starting with PROR: 2520
- Starting with PRS: 2520
- Starting with PRODIG: 360
- Starting with PRODIO: 720
- Starting with PRODIGI: 360

Adding these values gives us:

- Total = 90720 + 90720 + 181440 + 181440 + 10080 + 10080 + 20160 + 20160 + 10080 + 10080 + 2520 + 2520 + 2520 + 2520 + 2520 + 2520 + 360 + 720 + 360

Calculating this total gives us the rank of "PRODIGIOUS".

### Final Calculation
- Total = 90720 + 90720 + 181440 + 181440 + 10080 + 10080 + 20160 + 20160 + 10080 + 10080 + 2520 + 2520 + 2520 + 2520 + 2520 + 2520 + 360 + 720 + 360 = 609902

Thus, the rank of the word "PRODIGIOUS" is **609902**.

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