To diagnose why an engine does not start even though the battery is fully charged, it's crucial to check the electrical components and connections that are directly involved in the starting process. Let's explore each option provided and understand why "Loose battery terminal" is the correct next step to check:

1. Faulty ignition switch contacts:

   While a faulty ignition switch can cause starting issues, it usually would affect the starting mechanism more consistently, regardless of battery condition. Since the problem specified here occurs despite a fully charged battery, checking more immediate electrical connections is prudent first.

2. Blown main starter fuse:

   If the main starter fuse were blown, the starter wouldn't receive power to engage. However, this scenario would generally prevent any noise from the starter when attempting to start the vehicle. Checking this could come after initial checks of direct connections fail.

3. Loose battery terminal:

   This is the prime suspect here. Even with a fully charged battery, if the battery terminal connections are loose, the vehicle may not start. Loose terminals mean that the electrical current cannot effectively flow from the battery to the starter and other components necessary for starting the engine.

   Loose connections can be identified by manually checking the firmness of the terminal or by observing if there is play in the cable connections. Ensuring these are tight can often resolve the issue of a no-start condition for an engine with a charged battery.

4. Corroded solenoid switch connection:

   Corroded connections can also prevent the starter motor from engaging. However, similar to blown fuses, this would generally result in a different symptom pattern, such as intermittent starting. It's wise to check visible corrosion after ruling out the most common cause, which is the loose battery terminals.

In conclusion, the most immediate and likely cause of the engine not starting, despite a fully charged battery, is the "Loose battery terminal." Addressing the most straightforward physical connection issue should always come first, making this the ideal choice.