Here are some options for you to try:
1.check the captchpa number is correctly imputed
2. go to >Global Configuration and set the Captcha option to None… and, just to be sure go to User Manager>Options and disabled it there, too, try to register and see if this resolves
3. Generate a new key from Google and try that
4.
a. Open the FTP software installed on your computer
b. Browse to the /your web site address/templates/template-name/html/com_contact/ directory on your Joomla site
c.. Rename the /com_contact/ directory to /com_contact_old/
d. Joomla will now use the default contact form over ride files for the web site and the captcha form should now appear
**Note: Please replace the your web site address with your domain name and template-name with the template name you have downloaded***