A box contains 5 red, 4 white and 3 green balls. In how many ways can 3 balls…
2024
A box contains 5 red, 4 white and 3 green balls. In how many ways can 3 balls be drawn from the box, without replacement, so that at least 2 of them are green?
- A.
18
- B.
30
- C.
27
- D.
28
Show answer & explanation
Correct answer: D
CONCEPT: For an ‘at least’ condition on a draw without replacement, split the outcomes into mutually exclusive cases covering every count that satisfies the condition, count each case using combinations C(n, r) since order does not matter, then add the case totals (the addition principle).
Total balls = 5 red + 4 white + 3 green = 12, so green balls = 3 and non-green balls = 9.
Case 1 — exactly 2 green: choose 2 of the 3 green balls, C(3,2) = 3 ways, and 1 of the 9 non-green balls, C(9,1) = 9 ways, giving 3 × 9 = 27 ways.
Case 2 — exactly 3 green: choose all 3 green balls, C(3,3) = 1 way.
Since the two cases are mutually exclusive, add them: 27 + 1 = 28 ways.
CROSS-CHECK: Total ways to draw any 3 of the 12 balls = C(12,3) = 220. Ways with fewer than 2 green (0 or 1 green) = C(3,0) × C(9,3) + C(3,1) × C(9,2) = 84 + 108 = 192. Subtracting, 220 − 192 = 28, confirming the case-based total.