JotForm is a free online form builder which helps you create online forms without writing a single line of code. No sign-up required.
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It is possible to achieve your requirement. You can take advantage of our "Conditions" feature to show / hide form fields based on the input/selection of another form field.
Please take a look at the following cloned form where I have added the required condition to show / hide the drop down field based on the selection of the check boxes: http://form.jotformpro.com/form/41531103084946?
Feel free to clone this form for a closer look. The following guide should help you in form cloning: http://www.jotform.com/help/42-How-to-Clone-an-Existing-Form-from-a-URL
Hope this helps.
Do get back to us if you have any questions.