Answer
n(N $\cup$ U)=75
Work Step by Step
N$\cap$U is the set of authors who are new and unsuccessful.
N$\cup$U is the set of authors who are either new or successful (or both)
n(N) = 20 (total of column 1)
n(U) = 70 (total of row 2)
n(N$\cap$U)=15 (intersection of row 2 and column 1)
n(N$\cup$U)= (total new=20, which includes 15 unsuccessful) + (another 55 unsuccessful)
=75
(all 70 unsuccessful and all 20 new authors have been counted)
n(N)+n(U)-n(N$\cap$U)=20+70-15=75 (verified)