Suppose Anil's mother wants to give 5 whole fruits to Anil from a basket of 7 red apples, 5 white apples and 8 oranges. If in the selected 5 fruits, at least 2 oranges, at least one red apple and at least one white apple must be given, then the number of ways, Anil's mother can offer 5 fruits to Anil is
Show answer and solution
Answer: 6860
Solution
We need exactly 5 fruits with at least 2 oranges, at least 1 red apple, at least 1 white apple.
Let oranges = o, red apples = r, white apples = w where o+r+w = 5, o≥2, r≥1, w≥1.
Valid distributions (o, r, w):
- (2, 1, 2): C(8,2)·C(7,1)·C(5,2) = 28·7·10 = 1960
- (2, 2, 1): C(8,2)·C(7,2)·C(5,1) = 28·21·5 = 2940
- (3, 1, 1): C(8,3)·C(7,1)·C(5,1) = 56·7·5 = 1960
- (2, 3, 0): w=0, invalid
- (3, 2, 0): invalid
- (4, 1, 0): invalid
- (2, 0, 3): invalid
- (4, 0, 1): invalid
Wait, let me redo: o≥2, r≥1, w≥1, sum=5.
- (2,1,2): ✓ → 1960
- (2,2,1): ✓ → 2940
- (3,1,1): ✓ → 1960
- (2,3,0): w=0 ✗
- (3,2,0): ✗
- (4,1,0): ✗
Total =
Let oranges = o, red apples = r, white apples = w where o+r+w = 5, o≥2, r≥1, w≥1.
Valid distributions (o, r, w):
- (2, 1, 2): C(8,2)·C(7,1)·C(5,2) = 28·7·10 = 1960
- (2, 2, 1): C(8,2)·C(7,2)·C(5,1) = 28·21·5 = 2940
- (3, 1, 1): C(8,3)·C(7,1)·C(5,1) = 56·7·5 = 1960
- (2, 3, 0): w=0, invalid
- (3, 2, 0): invalid
- (4, 1, 0): invalid
- (2, 0, 3): invalid
- (4, 0, 1): invalid
Wait, let me redo: o≥2, r≥1, w≥1, sum=5.
- (2,1,2): ✓ → 1960
- (2,2,1): ✓ → 2940
- (3,1,1): ✓ → 1960
- (2,3,0): w=0 ✗
- (3,2,0): ✗
- (4,1,0): ✗
Total =
[JEE Main 2023 · 25 Jan · Shift 2]