Susan has $\mathbf{20}$ candies: $\mathbf{12}$ strawberry-flavored and $\mathbf{8}$ lemon-flavored. She gives Tony $\mathbf{4}$ candies. Tony, a math enthusiast, calculated the probability of getting exactly 2 strawberry and 2 lemon candies in a random selection of 4 candies.
(1) First, Tony calculated how many different selections of $4$ candies can be made from $20$. These are unordered quartets, and there are ${20\choose 4}=4\,845$ of them.
(2) In the next step, he calculated how many selections of $2$ strawberry candies can be made. He's choosing pairs from $12$ strawberry candies, and there are ${12\choose 2}=66$ such pairs.
(3) Next, Tony calculated the number of selections of $2$ lemon candies. He's choosing pairs from the total of $8$ lemon candies, and there are ${8\choose 2}=28$ such pairs.
(4) Tony found that he can choose $66$ pairs of strawberry candies and $28$ pairs of lemon candies. The total number of different quartets of candies containing $2$ strawberry and $2$ lemon candies is $66+28=94$.
(5) So, the probability of selecting exactly a quartet with $2$ strawberry and $2$ lemon candies from all the candies is $$\frac{94}{4\,845}\cong0.0194$$
Did Tony make a mistake in his reasoning? If so, find it!
Tony didn't make a mistake. He solved the task flawlessly, like a true math enthusiast!
He made an error right in step (1). The number of possible selections of $4$ candies from $20$ should have been calculated as an ordered quartet, which is $20\cdot19\cdot18\cdot17=116\, 280$. Therefore, the probability of getting exactly $2$ lemon and $2$ strawberry candies is: $$\frac{94}{116\,280}\cong0.0008$$
Tony made a mistake in steps (2) and (3) as well. When selecting pairs, the order of the candies matters. The number of pairs of strawberry candies should be $12\cdot11=132$, and the number of pairs of lemon candies should be $8\cdot7=56$. The correct probability is: $$\frac{188}{4\,845}\cong0.0388$$
Tony made a mistake in step (4). The number of different quartets of candies should be $66\cdot28=1\,848$, and the correct probability is: $$\frac{1\,848}{4\,845}\cong0.3814$$