Question: Of 128 boxes of oranges, each box contains at least 120 and at most 144 oranges. The number of boxes containing the same number of oranges is at least
Explanation:
Suppose we have five different types of fruits − A, B, C, D, E and seven boxes − 1, 2, 3, 4, 5, 6 and 7. We have to fill each box with only one kind of fruits.
To find the least number of boxes that will have same kind of fruit, we try to fill as many boxes as possible with different fruits. So we may have,
2: B
3: C
4: D
5: E
6: A
7: B
Therefore, there will be at least two boxes with the same fruits.
Applying a similar logic to the question, we try to fill the maximum number of boxes with different possible number of oranges.
Box 1: 120 oranges, Box 2: 121 oranges, …, Box 25: 144 oranges,
Box 26: 120 oranges, Box 27: 121 oranges, …, Box 50: 144 oranges,
Box 51: 120 oranges, Box 52: 121 oranges, …, Box 75: 144 oranges,
Box 76: 120 oranges, Box 77: 121 oranges, …, Box 100: 144 oranges,
Box 101: 120 oranges, Box 102: 121 oranges, …, Box 125: 144 oranges,
Box 126: 120 oranges, Box 127: 121 oranges, Box 128: 122 oranges
We can see that at least 6 boxes will have the same number of fruits.
Hence, option (c).