In what ways the letters of the word "PUZZLE" can be arranged to form the different new words so that the vowels always come together?

To solve this problem, we need to consider the word "PUZZLE" and its vowels and consonants. The word "PUZZLE" has 6 letters, with 2 vowels (U, E) and 4 consonants (P, Z, Z, L).

Step 1: Since the vowels need to always come together, we can consider the group of vowels (UE) as a single letter. So, we have 5 "letters" to arrange: P, Z, Z, L, and UE.

Step 2: We arrange these 5 "letters" in different ways. The formula for arranging n items (with repetition) is n!. However, since the letter Z is repeated, we divide by 2! to correct for over-counting. So, the number of arrangements is 5! / 2! = 60.

Step 3: However, within the "letter" UE, the U and E can be arranged in 2! = 2 ways.

Step 4: Therefore, the total number of ways the letters of the word "PUZZLE" can be arranged such that the vowels always come together is 60 * 2 = 120.