Whether or not full posts are displayed on Categories Pages and Archives pages is theme dependent ie. it's coded into the template and we cannot edit templates or themes. http://en.support.wordpress.com/themes/editing-themes/
In the majority of themes, posts on category or other archive pages show up just as they do on the main posts page in the blog and the number displayed is also the same. (See this page > Settings > Reading and note how many posts you choose there sets it for all). But in some themes only the beginnings of posts are shown, and in a few themes you only get titles or thumbnails. See here for a theme list of themes that may display the way you want > Category/archive pages: full posts or not http://wpbtips.wordpress.com/2009/03/25/full-posts-in-archive-pages/
As Twenty Ten does have an active except box you can do what houstonweaver is pointing to. On any theme with an active excerpt box you can use this workaround to have full posts on the front page.
You compose your whole post first and before you publish it you switch to the HTML (Text) editor, copy the code, paste it into the Excerpt box, and publish the post. http://en.support.wordpress.com/splitting-content/excerpts/#creating-excerpts
If you do not see an excerpt box then go to screen options on the top right hand corner of your Admin page and include:
Post – Likes and Shares, Format*, Categories, Tags, Excerpt, Send Trackbacks, Discussion, Comments**, Slug, Author, Revisions, Writing Helper
P.S. On themes that do have an active excerpt function, the workaround works very well but you have to do it on every post you create.