doing math

courses › Counting & Discrete Math

Counting

level 15 course

Multiply choices, arrange things in order, handle repeats.

Pen and paper is fine · no calculator needed why?

Learn first (about 3 minutes)

Opens at level 15.

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Builds on: Bigger multiplication (not open yet)

the lesson

The idea, the techniques and a tip for each skill, right here. The Learn page adds worked examples for every skill and untimed practice.

Read the lesson · about 3 minutes

The idea

Counting problems ask "how many ways", and most can be done without listing. Multiply the choices: 4 shirts and 3 pairs of trousers make 4 × 3 = 12 outfits, because every shirt goes with every pair.

Putting n different things in a row gives n × (n − 1) × … × 1 orders. That product is n factorial, written n followed by an exclamation mark. Picking only a few of them in order stops the product early. When some items are identical, swapping them makes no new arrangement, so you divide those swaps out.

Techniques

Multiply the choices

One pick from each of several groups.

  1. Count the options in each group.
  2. Multiply the counts. Every option in one group pairs with every option in the others.
worked example

Example: A café offers 3 starters, 5 main courses and 2 desserts. How many different meals with one of each can you order?

  1. 3 × 5 × 2 = 30.

Answer: 30

Fill the slots in order

Ordered picks, like first, second and third place, and factorial ratios.

  1. Count the choices for the first slot, then the next, one fewer each time.
  2. Multiply as many numbers as there are slots.
  3. Factorial ratios work the same way: 9 factorial ÷ 6 factorial cancels everything from 6 down, leaving 9 × 8 × 7.
worked example

Example: A club has 7 members. In how many ways can it choose a president, a vice-president and a secretary, if no one holds two posts?

  1. 7 choices for president, then 6, then 5.
  2. 7 × 6 × 5 = 210.

Answer: 210

Divide out the repeats

Arranging the letters of a word with repeated letters.

  1. Count all the orders as if every letter were different.
  2. For each repeated letter, divide by the factorial of how many times it appears.
worked example

Example: How many different arrangements are there of the letters in GEESE?

  1. 5 letters: 5 × 4 × 3 × 2 × 1 = 120 orders.
  2. The three Es swap in 3 × 2 × 1 = 6 ways.
  3. 120 ÷ 6 = 20.

Answer: 20

Tips by skill

  • TipMultiply the choices: Every option in one group pairs with every option in the others, so multiply the counts.
  • TipFactorial ratios: Everything from the smaller number down cancels. Multiply the numbers from the bigger one down to one above the smaller.
  • TipOrdered picks: Count the choices for each slot in turn, one fewer each time, and multiply.
  • TipArrangements with repeats: Count the orders as if all letters differ, then divide by the factorial of each repeated letter’s count.

Watch out for

  • Adding the options instead of multiplying. Every shirt goes with every pair of trousers.
  • Treating 10 factorial ÷ 8 factorial as 2 factorial. Cancel everything from 8 down; what is left is 10 × 9.
  • Dividing by the number of times a letter repeats instead of its factorial. Three Es means dividing by 6, not 3.

skills · practice stats

From rounds of this course only: box, review and test-out answers are left out. Once a skill has 40 tries, it compares your first 20 tries with your last 20.

rest ladder

Win 3 of your last 4 rounds and the course rests. A win is 90% right, within 2× the round's par. Pass the review when it comes back and the next rest is longer.

  1. 1 day
  2. 3 days
  3. 7 days
  4. 14 days
  5. 30 days
  6. 60 days
  7. mastered · every 90 days

your rounds

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