From a bag with 7 balls (2 red, 3 green, 2 blue), how many ways are there to draw two balls?

From a bag with 7 balls (2 red, 3 green, 2 blue), how many ways are there to draw two balls?

• A. 7
• B. 21
• C. 35
• D. 14

Answer: (B)

This question tests counting how many ways two items can be chosen when the order doesn’t matter. With seven balls total, the number of ways to draw two without regard to order is the combination 7 choose 2, which is 7×6/2 = 21. You can also see it by thinking in terms of color pair outcomes. The possible color pairs are RR, RG, RB, GG, GB, and BB. The number of distinct two-ball draws for each type are: RR (1 way), RG (2×3 = 6 ways), RB (2×2 = 4 ways), GG (C(3,2) = 3 ways), GB (3×2 = 6 ways), BB (1 way). Adding these up gives 1 + 6 + 4 + 3 + 6 + 1 = 21. If order mattered, there would be 7×6 = 42, but here we’re counting without regard to the order.

This question tests counting how many ways two items can be chosen when the order doesn’t matter. With seven balls total, the number of ways to draw two without regard to order is the combination 7 choose 2, which is 7×6/2 = 21.

You can also see it by thinking in terms of color pair outcomes. The possible color pairs are RR, RG, RB, GG, GB, and BB. The number of distinct two-ball draws for each type are: RR (1 way), RG (2×3 = 6 ways), RB (2×2 = 4 ways), GG (C(3,2) = 3 ways), GB (3×2 = 6 ways), BB (1 way). Adding these up gives 1 + 6 + 4 + 3 + 6 + 1 = 21. If order mattered, there would be 7×6 = 42, but here we’re counting without regard to the order.