Counting & Combinations
Fundamental counting principle, permutations, and combinations โ how many ways to arrange or choose โ excludes probability (see probability) and Venn diagram counting (see set-theory)
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Have you ever spent ages creating a custom character in a video game? ๐ฎ Imagine you can choose from 5 hairstyles, 4 shirts, and 3 pairs of shoes. How many totally unique characters can you make? You might be tempted to add the numbers, but the secret to unlocking all the possibilities is multiplying them! This is called the Fundamental Counting Principle, and itโs a superpower for the ISEE Quantitative Reasoning section.
Instead of drawing out every single outfit (which would take forever!), you just multiply: outfits! ๐คฏ
Sometimes the ISEE will ask you to arrange things, like putting books on a shelf or letters in a password. If order matters (like a password), it's called a permutation. If order doesn't matter (like picking two friends to share a pizza), it's a combination. For the ISEE, the most important trick is to draw blank lines for each choice you need to make, fill in the number of options for each blank, and multiply them together. Let's get counting! ๐
Practice Questions
3 practice questions for ISEE Lower Level
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- Use the fundamental counting principle: multiply the number of choices at each step. There are 3 choices of bread and 4 choices of filling. different sandwiches.
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- There are 3 morning choices and 2 afternoon choices. By the counting principle, the total number of combinations is .
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- For the first position there are 4 choices, for the second position 3 choices remain, for the third position 2 choices remain, and for the last position 1 choice. Multiply: .
Tips & Strategies
- Always check if repeats are allowed! If a question says 'no repeats' or 'without replacement', remember to subtract from your options for each new blank.
- Draw it out! Actually draw blank lines on your scratch paper for each choice (like _ \cdot _ \cdot _). It makes the math so much easier to see.
Common Mistakes
- Watch out for adding instead of multiplying! If you have 3 shirts and 4 pants, it's outfits, NOT .
- Don't forget to divide when order doesn't matter! If you are just picking a group of 2 people, picking 'Alex then Sam' is the same group as 'Sam then Alex'. You have to divide your total by to remove the duplicates!
Frequently Asked Questions
How do I know if order matters in a question?
Ask yourself if swapping the items makes a new thing. Swapping numbers in a password (123 vs 321) makes a new password, so order matters! Swapping flavors in a bowl of ice cream (chocolate and vanilla) is still the same bowl, so order doesn't matter.
What if I get completely stuck on a counting question?
Remember that there is NO penalty for guessing on the ISEE! If you're stuck, try to eliminate an answer that seems way too small (like if the numbers were added instead of multiplied), pick your favorite letter, and move on.
Do I need to memorize big formulas for combinations?
Nope! For the ISEE, you just need to know the 'draw the blanks and multiply' trick. If order doesn't matter, just divide by the number of ways to arrange the items you picked. Keep it simple!