...How to make an Applet using a Panel consisting FOUR different Layouts(FlowLayout, GridLayout...etc...) that shall make an output of 9 Buttons in a 3x3 layout...

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Is it just me, or does this smell like a homework assignment to anyone else?

You can only use one layout in a panel at a time. You can use multiple panels, nested, with different layouts, but a single Panel gets a single LayoutManager, as far as I know.

You can probably convince any LayoutManager to make a 3X3 grid - might be difficult with a BorderLayout, I guess. You just need to figure out which one makes it easiest to get closest to what you want, and which ones give you the flexibilities that you want.