Answer:
there are 144 permutations of the letters PRODUCT with consonants in the second and third positions.
Explanation:
The word PRODUCT has 6 letters, so there are 6! = 720 possible permutations of its letters.
To count the permutations where the second and third positions have consonants, we can first choose the consonants for these positions. There are 3 consonants in the word PRODUCT (P, R, and C), so we can choose 2 of them in C(3,2) = 3 ways. Once we have chosen the consonants, we can arrange them in the second and third positions in 2! = 2 ways (since there are two positions to fill).
The remaining 4 letters (O, U, and T) are vowels and can be arranged in the remaining 4 positions in 4! = 24 ways.
Therefore, the total number of permutations of the letters PRODUCT with consonants in the second and third positions is:
3 × 2! × 4! = 3 × 2 × 24 = 144
So there are 144 permutations of the letters PRODUCT with consonants in the second and third positions.