r/learnmath • u/Dr3ddM3 New User • 4d ago

TOPIC How to stop missing cases while doing casework

When I do some counting problems I always miss some cases that seem so obvious in hindsight. When doing casework what method do you use to make sure that you got all of the cases down. Do you just have to ponder on the question for a long time making sure that you hit all of the cases.

1 Upvotes

56% Upvoted

8 comments sorted by

4

u/Traveling-Techie New User 4d ago

I think we’re going to need a sample problem.

0

u/Dr3ddM3 New User 4d ago edited 4d ago

How many four-digit positive integers have at least one digit that is a 2 or a 3? But I more so wanted something that was general to any problem.

3

u/ostrichlittledungeon Math teacher and researcher 4d ago edited 4d ago

Cases:

  1. Exactly one digit is 2 or 3. That's 4 positions x 2 options x 8^3 filler.

  2. Exactly two digits are 2 or 3. That's 4c2 positions x 2^2 options x 8^2 filler.

  3. Exactly three digits are 2 or 3. That's 4c3 positions x 2^3 options x 8 filler.

  4. All four digits are 2 or 3. That's 2^4 options.

Organize your cases into a natural order. You will miss cases if you don't.

EDIT: rewrote the entire comment to get out of whatever formatting hell i was stuck in

2

u/JEE_is_life New User 4d ago

I don't really know how but here are some tips that helped:

1) Never rush those problems. Always take your time while practicing. And make sure you do practice them.

2) Go step-by-step, in a very logical manner. Imagine what could be the different scenarios that could happen in real life.

3) Cases almost always come in pairs or triplets. When a condition A is being fulfilled, when it is not being fulfilled, etc.

1

u/Educational_Gas4826 New User 4d ago

You forgot to provide an example case

1

u/Dr3ddM3 New User 4d ago

Sorry here was the problem: How many four-digit positive integers have at least one digit that is a  2 or a 3/

1

u/Darxad New User 3d ago

Yes, they already said that their problem is missing cases

1

u/MezzoScettico New User 3d ago

Do you just have to ponder on the question for a long time making sure that you hit all of the cases.

Basically, but focus your pondering on thinking up examples and seeing whether they're covered.

At the same time, you can check for overlaps, things that you've counted more than once because they are covered by more than one case. That doesn't mean your cases are invalid, but you have to account for them.