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← Unit 8 · Counting rules & factorials

Lesson

Read the rule, then work each example: try the setup yourself, then tap to check one step at a time.

The rules

Fundamental counting principle

Steps that build ONE outcome → multiply the number of choices: e₁ × e₂ × … × eₙ.

Can items repeat?

Repeats allowed (digits, PINs) → n × n × … = nʳ. No repeats (people, cards, letters of a WORD) → n × (n − 1) × (n − 2) …

Factorials

n! = n(n − 1)(n − 2)…1; arrange ALL n different things = n!; 0! = 1 and 1! = 1.

  1. Same n! on top and bottom? Cancel: 24!/22! = 24 × 23 = 552.
  2. 10! = 3,628,800 (exam Q22).
Traps and calculator keys are on the unit page →

Worked examples (9)

Example 1 · 2.4 #1a0/5

List all possible outfits

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Suppose you had 2 pairs of jeans (one black, one blue), and 3 T-shirts (one white, one blue, and one yellow). Find the possible outcomes for the outfits (one jean + one shirt)

Try the setup yourself first, then tap.

Example 2 · 2.4 #1b0/4

Use the formula to check your answer.

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Suppose you had 2 pairs of jeans (one black, one blue), and 3 T-shirts (one white, one blue, and one yellow). Find the possible outcomes for the outfits (one jean + one shirt)

Try the setup yourself first, then tap.

Example 3 · 2.4 #20/4

Suppose now you had 10 pairs of jeans, 15 T-shirts and 4 pairs of shoes. How many possible outfits could you choose from these?

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Try the setup yourself first, then tap.

Example 4 · 2.4 #30/4

Telephone numbers in the United States begin with three-digit area codes followed by seven-digit local telephone numbers. Area codes and local telephone numbers cannot begin with 0 or 1. How many different telephone numbers are possible?

[Figure: 10 boxes: 3 (area code) | 3 | 4 (local number); the 1st and the 4th box are tagged 'can not be 0 or 1']

Try the setup yourself first, then tap.

Example 5 · 2.4 #40/5

How many different phone numbers can be formed by the area codes 714 or 657 given that repetitions of the digits are not allowed?

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Try the setup yourself first, then tap.

Example 6 · 2.4 #5a0/4

$5!$

Find

Try the setup yourself first, then tap.

Example 7 · 2.4 #5b0/4

$(9 – 6)!$

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Find

Try the setup yourself first, then tap.

Example 8 · 2.4 #5c0/4

$\frac{24!}{22!}$

Find

Try the setup yourself first, then tap.

Example 9 · 2.4 #5d0/4

$\frac{24!}{21!3!}$

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Find

Try the setup yourself first, then tap.

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