A bag contains 2 red, 3 green and 2 blue balls. Two balls are drawn at random. What is the probability that none of the balls drawn is blue?

Explanation:-

$$\eqalign{ & {\text{Total}}\,{\text{number}}\,{\text{of}}\,{\text{balls}} \cr & = \left( {2 + 3 + 2} \right) = 7. \cr & {\text{Let}}\,{\text{S}}\,{\text{be}}\,{\text{the}}\,{\text{sample}}\,{\text{space}}{\text{.}} \cr & {\text{Then,}}\,n\left( S \right) = {\text{Number}}\,{\text{of}}\,{\text{ways}}\,{\text{of}}\, \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\text{drawing}}\,{\text{2}}\,{\text{balls}}\,{\text{out}}\,{\text{of}}\,{\text{7}} \cr & = {}^7{C_2} \cr & = \frac{{\left( {7 \times 6} \right)}}{{\left( {2 \times 1} \right)}} \cr & = 21. \cr & {\text{Let}}\,{\text{E = Event}}\,{\text{of}}\,{\text{drawing}}\,{\text{2}}\,{\text{balls,}}\, \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\text{none}}\,{\text{of}}\,{\text{which}}\,{\text{is}}\,{\text{blue}}{\text{.}} \cr & \therefore n\left( E \right) = {\text{Number}}\,{\text{of}}\,{\text{ways}}\,{\text{of}}\, \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\text{drawing}}\,{\text{2}}\,{\text{balls}}\,{\text{out}}\,{\text{of}}\,\left( {{\text{2 + 3}}} \right){\text{balls}} \cr & = {}^5{C_2} \cr & = \frac{{\left( {5 \times 4} \right)}}{{\left( {2 \times 1} \right)}} \cr & = 10. \cr & \therefore P\left( E \right) = \frac{{n\left( E \right)}}{{n\left( S \right)}} = \frac{{10}}{{21}}. \cr}$$