The best answer to this question is the second option. It is true that, depending on how one defines the words, the third option could be true, but the second option is better. While Booker T. Washington did not engage in anti-segregation efforts openly like WEB Du Bois or, later, Martin Luther King, Jr., he did engage in some efforts against segregation.
The first option is clearly not true. Washington is known for an accommodationist approach as laid out in his Atlanta Exposition speech, but Du Bois is not. This was a major difference between them.
The fourth option is clearly not true. The NAACP was not an organization that supported Washington’s accommodationist ideas. In addition, the link below shows that Washington was not one of the group that founded it.
That leaves the third option. Washington did not believe that African Americans were inherently inferior to whites. That means that we could say that he “supported full racial equality.” However, I take this statement to mean that he wanted African Americans to achieve full racial equality with whites in his own time. This is clearly not true. Washington thought that it was a bad idea to press for full racial equality in the short term.
If the other options are wrong, then the second option must be right. Washington did not engage in protests against segregation or anything like that. However, he did help to fund lawsuits that attempted to break down the system of segregation. I interpret this as activism against segregation. Therefore, I think the second option is best.
It might be a good idea for you to check your text or notes to see if they interpret the second and third options differently than I do. If they do, the third option might be correct. However, it is my view, based on my understanding of the two statements’ meanings, that the second option is better.