Couldn't we do casework with all different and two of one number?

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[Originally posted in Discussions]
Module 0 Week 1 Day 3 Something to Think About
Yes, you could! The reason Dr. Loh doesn't focus on this way is because you might have thought of that already, and he wants to show you a neat, different way of doing it.
The casework method would be as follows:
Case 1: All different digits
The digits have to be 1, 2, and 3, and there are 3! = 6 ways to mix them up to create a threedigit number.
Case 2: Two same digits, the third different
There are three possibilities for the doubledigit: 1, 2 or 3. After we have chosen the doubledigit, we have two choices for the other, different digit. This gives \(3 \times 2 = 6\) ways. However, we can mix them up. For example, 122 can be mixed up in three ways \((122, 212, \text{ and } 221).\) So we have \( 6 \times 3 = 18\) ways for this case.
Total:
Ways from Case 1 + Ways from Case 2 =
$$ 6 + 18 = 24 \text{ total ways } $$There's an urge in mathaficionados and mathphiles to always search for new, clever or surprising ways of solving problems. It's the same urge that makes a computer programmer spend hours writing a program to do a mundane calculation that might take him the same amount of time if he did it by hand. And it's the same urge that makes you want to tell a little kid to count his legos by making piles of 10, rather than counting one by one.
If you still have any more questions, please don't hesitate to ask!
Happy Learning,
The Daily Challenge Team