Counting Principles Flashcards

All 9 cards in this deck

What is the product rule for counting?

Multiply together the number of choices at each stage.

e.g. 3 choices then 4 choices gives $3 \times 4 = 12$ outcomes.

What are the steps to count arrangements when no repetition is allowed?

e.g. how many 3-digit numbers can be made from 1, 2, 7, 8, 9 using each digit at most once?

1. Count the choices for the first position: 5.
2. Each later position has one fewer choice: 4, then 3.
3. Multiply: $5 \times 4 \times 3 = 60$.

What are the steps to count outcomes with two restrictions that clash, by splitting into cases?

e.g. 5-digit numbers using 1, 2, 7, 8, 9 each once, greater than 40 000 and odd

1. Split into cases on the clashing position: first digit 8, 7 or 9.
2. Use the product rule in each case: 18, 12, 12.
3. Add the case totals: $18 + 12 + 12 = 42$.

True or false? A digit that must be greater than 6 has 4 possible values.

False. It has 3 values (7, 8, 9) — 'greater than' does not include the boundary digit 6.

True or false? A digit that must be less than 8 has 8 possible values.

True. They are 0, 1, 2, 3, 4, 5, 6, 7 — zero must not be left out.

What are the steps to count arrangements when repetition is allowed?

e.g. how many 3-digit codes can be made from the digits 0–9 if digits may repeat?

1. Count the choices for each position separately: 10 for each of the 3 positions
2. Note the choices do not reduce, because digits may be reused
3. Multiply: $10 \times 10 \times 10 = 1000$ codes

What are the steps to count numbers with a restriction on the final digit?

e.g. 3-digit numbers made from 1, 2, 3, 4, 5 with no repetition that are odd

1. Fill the restricted position first: last digit odd $\rightarrow$ 3 choices (1, 3, 5)
2. Count the choices left for the other positions: 4 then 3
3. Multiply: $3 \times 4 \times 3 = 36$ numbers

In a counting problem, what must the final digit be for the number to be a multiple of 5?

0 or 5 — so 2 choices, if both are available digits.

True or false? When counting even numbers, 0 counts as a possible even final digit.

True.
The even final digits are 0, 2, 4, 6, 8 — wrongly excluding zero loses marks.