Mathematical Reasoning

means means
means

Correct result is as follows:

or

Step 1: Identifying the letters
The word "PERMUTATION" consists of 11 letters: The vowels are (5 vowels), and the consonants are (6 consonants).
Step 2: Arranging the consonants first
The 6 consonants (PRMTTN) are arranged, considering the repetition of T:
Step 3: Placing vowels in available slots
Since vowels should not be adjacent, they must be placed in the 7 available gaps among the consonants. The number of ways to choose 5 out of 7 slots:
The 5 vowels can be arranged among themselves:
Step 4: Eliminating cases where T’s are together
Among these permutations, the number of cases where both T’s are together:
Step 5: Final Calculation Final Answer: The correct answer is .