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Counting

Podiums and committees, the letters of BANANA, a round table, "at least one", sweets shared out: count exactly, and know which formula and why.

  • Quantitative reasoning
  • Untimed
  • Endless levels

How it plays

Each question asks how many ways something can happen. First decide whether order matters (a podium) or not (a team), whether things repeat (the three A's in BANANA), and whether the arrangement turns (a round table).

Later levels bring the counts that catch people: "at least one" (count the groups with none, and subtract), two friends who must sit together (glue them into one block), and sharing identical things out (stars and bars), with and without a cap.

Every answer is exact, and each formula is checked against counting every case one by one. The wrong options are the real mistakes: order counted when it does not matter, repeats ignored, "pick one senior, then any" double counting.

Keys

Space starts and skips ahead · 1–9 pick · Enter submits · Esc stops. On a phone, tap.

What it trains

This makes counting arrangements and selections of these kinds quicker and surer. It claims nothing wider.

Learn

How to get better at counting: the method, the techniques, what each new level asks of you, and the traps the wrong answers are made of.

Your approach to each question

  1. Ask three questions before any arithmetic. Does order matter — a podium, or a team? Do things repeat — the three A’s in BANANA, or identical sweets? Does the arrangement turn — a row, or a round table?
  2. Look for the awkward condition next: “at least one”, “next to each other”, “not next to each other”, “each at least one”, “at most so many”. Each has its own standard move, below.
  3. Write the formula with the numbers in before you multiply anything out, and cancel first. 8 × 7 × 6 ÷ (3 × 2 × 1) is 8 × 7 = 56 once the 6 cancels.
  4. With options shown, find the mistakes among them: one is the ordered count, one ignores repeats, one double counts. Knowing which slip made each option tells you which one is left.

Techniques and tricks

  • Order matters: count down

    For k places filled from n, multiply n × (n − 1) × … for k factors. Each place has one fewer choice than the one before.

    e.g. 8 runners, gold, silver, bronze: 8 × 7 × 6 = 336

  • Order does not: divide by k!

    A team of k is the ordered count divided by k!, because each team was counted once for every order of its members. Cancel before you multiply. Choosing k is the same as leaving out n − k, so use whichever is smaller.

    e.g. C(8, 3) = 336 ÷ 6 = 56; C(10, 6) = C(10, 4) = 210

  • Repeats: divide by each repeat’s factorial

    Arrange all the letters as if different, n!, then divide by m! for every letter that appears m times, since swapping identical letters changes nothing.

    e.g. BANANA: 6! ÷ (3! × 2!) = 720 ÷ 12 = 60

  • Round table: fix one seat

    Rotations count as the same, so seat one person anywhere, then arrange the other n − 1 in a row: (n − 1)!.

    e.g. 5 people round a table: 4! = 24

  • Glue the pair into a block

    For two who must sit together, treat them as one block: (n − 1)! orders, times 2 for the order inside the block. For “not together”, subtract that from n!.

    e.g. 5 in a row, Asha and Ben together: 2 × 4! = 48; apart: 120 − 48 = 72

  • At least one: count none, and subtract

    Count all the groups, then the groups with none of the thing, and subtract. Never “pick one, then any”: that counts a group with two of them twice.

    e.g. 3 from 5 juniors and 4 seniors, at least one senior: C(9, 3) − C(5, 3) = 84 − 10 = 74

  • Stars and bars, then the cap

    n identical things among k people is C(n + k − 1, k − 1); with “each at least one”, hand out one each first and share the rest. For a cap, subtract the solutions where one variable is over it, then add back those where two are.

    e.g. x + y + z = 10, each 0 to 4: 66 − 3 × 21 + 3 × 1 = 6

As the levels climb

  • The kinds unlock in this order: order matters and order does not; then repeated letters and round tables; then two who sit together and “at least one”; then sharing out with stars and bars; then sharing with a cap. Four in ten questions come from the newest kind.
  • The numbers grow as you climb — more runners, bigger groups, longer words up to MISSISSIPPI. The method does not change, but the arithmetic does: cancel the factorials before multiplying, and keep 10! = 3,628,800 in mind as a sense of size.
  • Each step of size adds 8% to the time, up to twelve steps, so longer arithmetic gets more of it. Spend it on deciding the formula; the multiplying is the easy part once the cancelling is done.
  • The cap questions at the top need inclusion–exclusion in full: count without the cap, take away each variable over it, add back each pair over it. If two variables cannot both be over the cap, the last step is zero.

Traps to avoid

  • Order counted when it does not matter: 336 given for a team of 3 from 8. A team is the same team in any order; divide by 3! to get 56.
  • Repeats ignored, or only one of them divided out: 720 given for BANANA, or 720 ÷ 3! = 120. Every repeated letter needs its own factorial — the A’s and the N’s.
  • n! for a round table, or (n − 1)! ÷ 2: this game counts only rotations as the same, not mirror images, so it is (n − 1)! exactly.
  • “Pick one senior, then any of the rest”: 4 × C(8, 2) = 112, more than the 84 groups there are at all. It counts a group with two seniors twice. Use all minus none.
  • kⁿ for identical things: that counts which person gets each different item. With identical sweets only the heap sizes matter, so it is stars and bars.

How to practise

  • Use “Kinds of question” to pair the two that get confused — “Order matters” with “Order does not matter” — until you decide which it is before reading the numbers.
  • Switch “Answer by” to “Typing the answer”. You have to give the exact count, so the arithmetic must be finished, and cancelling becomes a habit.
  • Raise “Options shown” towards 8 to see more of the classic wrong counts at once, and name the mistake behind each before you choose.
  • Read the working on every miss. Each one says why its formula is right in a line, and that line is what to remember next time.

Levels

It is untimed unless you turn the clock on. There is no last level: at least one dial turns without end. Your level is judged on your last ten answers, every ten: 90% moves you up, below 80% moves you down, which holds most people near four in five right.

All of it is in the settings beside Start — how many answers a judgement looks at and how often it recalibrates, the thresholds, the time, which questions, how many options, and whether you see each verdict at once or review them together at the judgement. You can also jump to any level.

More quantitative reasoning

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