Here's how to determine the number of ¹³C-NMR signals for acetylferrocene:

1. Understand the Structure

Acetylferrocene has the following structure:

* Ferrocene Core: Two cyclopentadienyl rings (Cp) bound to an iron atom (Fe).

* Acetyl Group: A CH3CO- group attached to one of the Cp rings.

2. Identify Equivalent Carbons

* Cp Rings: Due to the rotational symmetry of the ferrocene core, the carbons on each Cp ring are equivalent.

* Acetyl Group: The carbons in the acetyl group are distinct.

3. Count the Signals

* Ferrocene Cp Rings: 5 carbons per Cp ring, but equivalent, so only 1 signal from each Cp ring.

* Acetyl Group: 2 distinct carbons (CH3 and CO).

Therefore, you would expect to see a total of 4 signals in the ¹³C-NMR spectrum of acetylferrocene.