The word HAPPINESS consists of the letters H, A, P, P, I, N, E, S, S. We need to select 4 letters with no Ps and either one or two Ss.

First, remove the Ps, leaving us with the letters H, A, I, N, E, S, S.

Case 1: One S

Select 1 S and 3 other letters from H, A, I, N, E. This can be done in \(\binom{5}{3} = 10\) ways.

Case 2: Two Ss

Select 2 Ss and 2 other letters from H, A, I, N, E. This can be done in \(\binom{5}{2} = 10\) ways.

\(Total number of selections = 10 + 10 = 20.\)

Alternatively, select 3 letters from H, A, I, N, E, S (6 letters total) and ensure at least one S is included. This can be done in \(\binom{6}{3} = 20\) ways.