Multiple choice

A box contains 10 balls out of which 3 are red and the rest are blue. In how many ways can a random sample of 6 balls be drawn from the bag so that at the most 2 red balls are included in the sample and no sample has all the 6 balls of the same colour?

  1. 105

  2. 168

  3. 189

  4. 120
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Total balls = 10 (3 red, 7 blue). We need a sample of 6 with at most 2 red balls. Cases: 0 red (7C6 = 7), 1 red (3C1 * 7C5 = 3 * 21 = 63), 2 red (3C2 * 7C4 = 3 * 35 = 105). Total = 7 + 63 + 105 = 175. We must exclude samples with 6 balls of the same color. Only 6 blue balls is possible (7C6 = 7). 175 - 7 = 168.

AI explanation

The box contains 3 red and 7 blue balls. A sample of 6 balls with at most 2 red balls can be chosen in (3C1 x 7C5) + (3C2 x 7C4) ways. Evaluating this gives (3 x 21) + (3 x 35) = 63 + 105 = 168. The total selections are 168.

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