Anagrams containing the block GIO

Text

How many anagrams of the word GIOCARE (whose $7$ letters are all different) contain the block “GIO”, with these three letters consecutive and in that order?

Solution

We treat the block “GIO” as a single glued element. Then $5$ all-distinct elements remain to be arranged: the block $[\text{GIO}]$ and the letters $C, A, R, E$. The number of orderings is $N = 5! = \boxed{120}.$