Final answer:
The number of unique permutations of the letters in the word QUANTITATIVE is found by calculating 12! and then dividing by the factorials of the frequencies of the repeating letters. The final answer is 19,958,400 unique permutations.
Step-by-step explanation:
The question asks us to find the number of unique permutations of the letters in the word QUANTITATIVE. To solve this, we use the formula for permutations of a word with duplicate letters, which is n! divided by the factorial of each uniquely repeating letter's frequency. In this case, the word QUANTITATIVE has 12 letters. The letter T occurs 3 times, and the letter A and I each occur twice. Therefore, the total number of unique permutations is 12! / (3! * 2! * 2!)
Let's compute this step by step:
- First, we calculate 12!, which is 479,001,600.
- Then we calculate the factorial for the repeating letters: 3! for the T's (which is 6), and 2! for both A's and I's (each is 2).
- We then multiply the factorials of the repeating letters to get the divisor, which is 3! * 2! * 2! = 6 * 2 * 2 = 24.
- Finally, we divide the total number of permutations by the divisor: 479,001,600 / 24 = 19,958,400 unique permutations.
So, there are 19,958,400 unique permutations for the letters in the word QUANTITATIVE.