We continue with our counting lessons — and once again, Kitten proves that she doesn’t think the same way I do. In fact, her solution is so elegant that I think she could have a future as a mathematician. After all, every aspiring novelist needs a day job, right?
If only I could get her to give up the idea that she hates math…
Permutations with Complications
How many of the possible distinct arrangements of 1-6 have 1 to the left of 2?
My Method: Hit the Problem with a Sledgehammer
It took some time, but I did get the answer. I used casework — that is, I thought of all the possible things that might happen to satisfy the requirements, counted them individually, and added up my possibilities to get the final answer.
The cases I considered were all the places that the 1 could go and still have the 2 to its right. Making a 6-digit number, we could have:
1 _ _ _ _ _
_ 1 _ _ _ _
_ _ 1 _ _ _
_ _ _ 1 _ _
_ _ _ _ 1 2
Then I filled in the number of choices available for each blank space, using the Fundamental Counting Principle and taking into account that the 2 has to come after the 1. After that, I multiplied everything out (five multiplications, one for each line). And finally, I added those products together to find my answer.
Ugly, but it works.
The Book’s Method: Flash of Insight Required
I didn’t like the solution in the book because it requires a flash of insight. Such “Aha!” moments are fun but notoriously unreliable. The book used symmetry to solve the problem, recognizing that every possible arrangement of digits, such as:
1 2 3 4 5 6
has a mirror-image partner:
6 5 4 3 2 1
And in every such mirror-image pair, only one of them will have the 1 and 2 in the correct order. So we can just count the number of possible arrangements of digits and then divide by 2.
Nice, if you happen to think of it.
Kitten’s Method: Build the Number, Counting As You Go
Kitten’s method only took two short lines on her whiteboard — though it will take much longer here, since I have to type out her explanation. To begin with, the 1 and 2 must be in this order:
Choices = 1. Total possible numbers so far:
1 (the choices for the other digits)
I almost interrupted her to say that the 1 and 2 don’t have to sit next to each other. Thankfully, I kept my mouth shut, and thus saved myself from sticking my foot in it. She had everything under control.
The next digit can fit before the 1, or after the 2, or in between them:
_ 1 _ 2 _
Choices = 3. Total possible numbers so far:
1 3 (the choices for the other digits)
Kitten put a 3 at the end for demonstration purposes, but as she explained, the possibilities are the same no matter which digit we use or where we put it.
The next digit has 4 choices of where to go:
_ 1 _ 2 _ 3 _
1 3 4 (the choices for the other digits)
The next digit has 5 choices of where to go:
_ 1 _ 2 _ 3 _ 4 _
1 3 4 5 (the choices for the last digit)
The final digit has 6 choices of where to go:
_ 1 _ 2 _ 3 _ 4 _ 5 _
Total possible numbers that fit our criterion:
1 3 4 5 6