Counting & Combinations
Fundamental counting principle, permutations, and combinations — how many ways to arrange or choose — excludes probability (see probability) and Venn diagram counting (see set-theory)
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Imagine you're at an ice cream shop. You have 3 flavors, 2 syrups, and 4 toppings. How many different sundaes can you make? 🍦 Counting isn't just going 1, 2, 3... It's about figuring out how many choices or combinations you have! This is called the Fundamental Counting Principle. If you multiply your choices together, you get the total number of combinations.
Think about getting dressed. If you have 3 cool t-shirts and 2 pairs of shorts, you just multiply to find out you have 6 different outfits! đź‘•đź‘–
On the SSAT, they love asking you to count things. Sometimes it's outfits, sometimes it's pizza toppings, and sometimes it's figuring out how many boxes of something you need to buy. The trick is to read carefully and see if you need to multiply your choices, or just use basic division to see how things fit into groups!
Sometimes order matters, like guessing a secret password. Other times, order doesn't matter, like picking which 2 friends to invite to the movies. 🍿 For the SSAT, especially at the lower and middle levels, you'll mostly use basic multiplication to find total combinations, or division to group items together. Just take it one step at a time, draw a quick picture if it helps, and you'll be a counting master in no time!
Practice Questions
3 practice questions for SSAT Upper Level
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- To solve this, treat the 2 parents as a single "block" or unit since they must remain together. This means we are arranging 5 units in total (the 4 individual children plus the 1 parent block). The number of ways to arrange these 5 units in a row is .
Within their block, the 2 parents can also swap places with each other, which can be done in ways.
Multiply the number of arrangements of the units by the number of arrangements within the block to find the total number of ways the family can line up: .
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- First, find the number of ways to select 1 boy from the 4 available boys. This is simply ways.
Next, find the number of ways to select 2 girls from the 5 available girls. Since the order of selection does not matter, use the combination formula: ways.
To find the total number of possible groups, multiply the number of ways to choose the boy by the number of ways to choose the girls: different groups.
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- We can systematically list the possible color combinations based on how many colors are chosen.
Case 1: All 3 pens are the same color.
Since there are 4 red pens, we can choose 3 red pens (Red-Red-Red). We cannot choose 3 blue or 3 black pens because there are only 2 of each in the drawer. (1 combination)
Case 2: 2 pens of one color and 1 pen of another color.
- With 2 reds, the third pen can be blue or black (Red-Red-Blue, Red-Red-Black).
- With 2 blues, the third pen can be red or black (Blue-Blue-Red, Blue-Blue-Black).
- With 2 blacks, the third pen can be red or blue (Black-Black-Red, Black-Black-Blue).
(6 combinations)
Case 3: All 3 pens are different colors.
We must select exactly 1 red, 1 blue, and 1 black pen (Red-Blue-Black). (1 combination)
Total possible color combinations = .
Tips & Strategies
- When in doubt, draw a tree diagram! If the question asks about shirts and pants combinations, draw branches for each choice. The number of endpoints is your total count. 🌳
- Look for the keyword 'each.' If a question says 'for each appetizer, pick one main course,' that's a signal to MULTIPLY the choices together!
Common Mistakes
- Don't add when you should multiply! If you have 3 shirts and 4 pants, the answer is outfits, NOT . Adding gives you the total items, not the total combinations!
- Watch out for whether repetition is allowed! A 3-digit password where digits CAN repeat has options. If digits CANNOT repeat, it's .
Frequently Asked Questions
Do I need to memorize complicated combination formulas for the SSAT?
Nope! 🧠For the lower and middle level SSAT, you mostly just need to use the Fundamental Counting Principle. Just multiply your choices together!
What if a counting question has a ton of choices?
Break it down step by step! Write down how many options you have at each decision point, then multiply them all together at the end.
What's the difference between a combination and a permutation?
In a permutation, ORDER matters (like a password — 123 is different from 321). In a combination, order does NOT matter (like picking 2 friends for a team). For the SSAT, the Fundamental Counting Principle covers most questions!
Will they ask me about probability too?
Yes! Probability is just counting the winning choices divided by the total choices. So if you can count your combinations, you're already halfway to solving probability questions! 🎲