Question

Boxes numbered 1, 2, 3, 4 and 5 are kept in a row and they are necessarily to be filled with either a red or a blue ball such that no two adjacent boxes can be filled with blue balls. How many different arrangements are possible, given that the balls of a given colour are exactly identical in all respects ?

• A. $8$
• B. $10$
• C. $13$
• D. $22$

Answer 1

The correct option is C $13$

Make cases when all $5$ boxes are filled by:

Case 1: Identical $5$ red balls ${=}^5{C}_5=1$ way

Case 2: $4$ identical red balls and $1$ blue ball ${}^5{C}_1=5$ ways

Case 3: $3$ red balls and $2$ blue balls

Is similar to filling $4$ gaps by $2$ blue balls ${=}^4{C}_2=6$ ways

Case 4: $2$ red and $3$ blue balls

Is similar to filling $3$ gaps by $3$ blue balls ${=}^3{C}_3=1$ way

$\therefore$ Total number of ways $=1+5+6+1=13$ ways