You can allow students to attach files to a Discussion by changing the course settings. Settings are available in the Discussions Page or the Course Settings Page.

Discussions Page Settings

In Course Navigation, click the Discussions link.

Open Settings Icon

Click the Settings icon to open Discussion settings.

Edit Discussion Settings

Make sure you mark the Attach files to discussions checkbox.

Note: Unchecking this option will restrict students from attaching files within Discussions. This permission affects posts in new discussion topics, discussion replies, and discussions within course groups.