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Junior Kangaroo 2024 Solutions
16. A bag contains 3 green apples, 5 yellow apples, 7 green pears and 2 yellow pears. George takes out pieces of fruit at random, one piece at a time. How many pieces of fruit must he take to be certain he has at least one apple and one pear of the same colour?
Options:
A 9
B 10
C 11
D 12
E 13
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Let's understand this problem step by step. We have a bag containing 3 green apples, 5 yellow apples, 7 green pears, and 2 yellow pears. George draws fruit randomly one at a time. We need to find the minimum number of pieces he must draw to guarantee he has at least one apple and one pear of the same color. This means either a green apple with a green pear, or a yellow apple with a yellow pear.
To solve this problem, we need to think about the worst-case scenario. What's the maximum number of fruits George can draw without getting at least one apple and one pear of the same color? We want to avoid having both a green apple and green pear, and also avoid having both a yellow apple and yellow pear. Let's analyze the different ways George can draw fruits while avoiding these matching pairs.
Let's examine all four possible worst-case scenarios. Case 1: George draws only pears, avoiding all apples. This gives him 9 fruits maximum. Case 2: He draws no green apples and no yellow pears, getting 5 yellow apples and 7 green pears for 12 fruits total. Case 3: He draws no green pears and no yellow apples, getting only 5 fruits. Case 4: He draws only apples, avoiding all pears, for 8 fruits maximum. The worst case is Case 2 with 12 fruits, where George has yellow apples and green pears but no matching color pairs.
Now let's see what happens after the worst-case scenario of 12 draws. George has drawn all 5 yellow apples and all 7 green pears, but he has no matching color pairs yet. What remains in the bag? Only 3 green apples and 2 yellow pears. When George draws the 13th fruit, it must be either a green apple or a yellow pear. If it's a green apple, he now has both green apples and green pears, creating a green pair. If it's a yellow pear, he now has both yellow apples and yellow pears, creating a yellow pair. Either way, the 13th draw guarantees success!
Let's summarize our solution. We found that the maximum number of fruits George can draw without getting a matching color pair is 12, achieved by drawing all 5 yellow apples and all 7 green pears. After these 12 draws, only 3 green apples and 2 yellow pears remain. The 13th draw must be either a green apple, which creates a green pair, or a yellow pear, which creates a yellow pair. Either way, success is guaranteed on the 13th draw. Therefore, the answer is 13, which corresponds to option E.