A teacher reaches into a bag of game pieces containing 5 black pieces, 3 purple pieces, 1 orange piece, and 6 green pieces. He draws 2 pieces, setting aside the first piece before drawing the second. What is the probability that at least one of the drawn pieces is purple

The probability of choosing a purple piece is 3/15, choosing an orange piece will be 1/15, that of black pieces is 5/15 and that of choosing a green piece is 6/15.
This would be PP + PW+ PG + PB +WP +GP + BP
                   = (3/15 × 2/14) + (3/15×1/14)+(3/15×6/14) +(3/15+5/14)+(1/15×3/14) + (6/15× 3/14) + (5/15 × 3/14)
                     = 6/210 +3/210 +18/210+15/210+3/210 +18/210+ 15/210 
                     = 78/210
                     = 13/35