9C0+9C1+......+9C9 gives all possible selections. In these sections we look for selections with at least one red ball. r=1 section R is in 1 way r=2 sections RR , RB are in 2 way. r=3 sections RRR , RRB, RBB are in 3 ways. r=4 sections RRRR , RRRB , RRBB, RBBB are in 4 ways r=5 sections RRRRB , RRRBB, RRBBB, RBBBB are in 4 ways r=6 sections RRRRBB , RRRBBB, RRBBBB, RBBBBB are in 4 ways. r=7 sections RRRRBBB, RRRBBBB , RRBBBBB are in 3ways. r=8 sections RRRRBBBB , RRRBBBBB are in 2 ways r=9 section of all balls is in 1 way Total=24 ways of selecting at least one red ball.