In a form, I have a field called mode of transport: for that I used three radio buttons like own , private and public.
If he clicks public radio button, it should show two radio buttons like pickup and drop. Otherwise it should not show it(pickup and drop). I don`t know how to do? can anyone tell it with simple example

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You can attach a click event to the radiobuttons to show/hide the pickup/drop radios. This will be done in javascript, so nothing to do with PHP, even if you create the form via PHP. Moving to JS forum...