The first answer I gave you was incorrect. I regret this.
Please ignore it and accept my apology. Here is the correct
solution:
We will first find the probability of the complement event,
where there are only 1 or 2 types of toys. Then we will
subtract from 1 to find the desired probability.
Let the 5 purchases be purchases 1,2,3,4, and 5, say in
order of the earliest purchase to the latest purschase.
Case 1. 3 purchases getting one kind of toy and 2 purchases
getting another kind of toy.
Case 2. 4 purchases getting one kind of toy and 1 purchase
getting another kind of toy.
Case 3. 5 purchases getting the same kind of toy all 5 times.
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Now we calculate the number in each of the above 3 cases.
Case 1. 3 purchases getting one kind of toy and 2 purchases
getting a second kind of toy.
Choose the 3 purchases for getting one kind of toy.
That's 5 purchases choose 3 or 5C3 ways.
Choose the remaining 2 purchases for getting the
second kind of toy.
That's 2 purchases choose 2 or 2C2 ways.
Choose the type toy for the three of one kind 3C1 ways
Choose the type toy for the two of another kind in 2C1 ways
For case 1, 5C3×2C2×3C1×2C1 = 10×1×3×2 = 60 ways
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Case 2. 4 purchases getting one kind of toy and 1 purchase
getting another kind of toy
Choose the 4 purchases for getting one kind of toy.
That's 5 purchases choose 4 or 5C4 ways.
Choose the remaining 1 purchase for getting the
other kind of toy.
That's 1 purchase choose 1 or 1C1 ways.
Choose the kind of toy for the four purchases getting the
same kind of toy.
That's 3 toy types choose 1 or 3C1
Choose the toy type for the two purchases getting the
other kind of toy.
That's 2 toy types choose 1 or 2C1
For case 2, 5C4×1C1×3C1×2C1 = 5×1×3×2 = 30 ways
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Case 3. 5 purchases getting the same kind of one toy
in each purchase.
That's 3 toy types choose 1, or 3C1
For case 3, 3C1 = 3
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Total for all three cases: 60+30+3 = 93
So the numerator for the probability of the complement
event is 93
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Now we calculate the denominator for the probability.
That's all the ways the toy types can come up.
Choose the toy type to get in the 1st purchase in 3 ways.
Choose the toy type to get in the 2nd purchase also in 3 ways.
Choose the toy type to get in the 3rd purchase also in 3 ways.
Choose the toy type to get in the 4th purchase also in 3 ways.
Choose the toy type to get in the 5th purchase also in 3 ways.
That's 3×3×3×3×3 = 35 = 243 ways.
So the probability of the complement event is 93/243 = 31/81.
Therefore the probability of the desired event is
1 - 31/81 = 81/81 - 31/81 = 50/81
THAT's the correct answer. Sorry I got it wrong the first time.
Edwin