Consider 4 boxes, where each box contains 3 red balls and 2 blue balls. Assume that all 20 balls are distinct. In how many different ways can 10 balls be chosen from these 4 boxes so that from each box at least one red ball and one blue ball are chosen?
- (A)(A) 21816✓ Correct
- (B)(B) 85536
- (C)(C) 12096
- (D)(D) 156816
Answer: (A)
Solution
We need to select 10 balls from 4 boxes such that each box contributes at least 1 red and 1 blue ball.
Total balls needed: 10. Minimum from each box: 2 (1R + 1B). So 8 balls are fixed, and we need to select 2 more.
Case I: One box provides 4 balls (either 3R+1B or 2R+2B), and the other three boxes provide 2 balls each (1R+1B).
Number of ways:
Total:
Case II: Two boxes provide 3 balls each (either 2R+1B or 1R+2B), and the other two boxes provide 2 balls each (1R+1B).
Number of ways:
Total:
Total ways:
Therefore, the answer is (A).
Total balls needed: 10. Minimum from each box: 2 (1R + 1B). So 8 balls are fixed, and we need to select 2 more.
Case I: One box provides 4 balls (either 3R+1B or 2R+2B), and the other three boxes provide 2 balls each (1R+1B).
Number of ways:
- Choose which box provides 4 balls: 4 ways
- From that box, choose 4 balls with at least 1R and 1B: ways
- From each of the other 3 boxes, choose 1R and 1B: ways
Total:
Case II: Two boxes provide 3 balls each (either 2R+1B or 1R+2B), and the other two boxes provide 2 balls each (1R+1B).
Number of ways:
- Choose which 2 boxes provide 3 balls: ways
- From each of these 2 boxes, choose 3 balls with at least 1R and 1B: ways (excluding all same color)
- From each of the other 2 boxes, choose 1R and 1B: ways
Total:
Total ways:
Therefore, the answer is (A).
JEE Advanced · 2022 · 28 Aug 2022 · Paper 2 · Single correct option