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← Introduction to Counting and Probability levels

Introduction to Counting and Probability / LEVEL 1 · DIFFICULTY 1/5

Choices and Arrangements

Decide when order and repetition matter.

3 stages · 24 practice problems · two 6-question assessment forms

STAGE CP 1.1 · 8 PRACTICE PROBLEMS · 2 NEW-CONTEXT APPLICATIONS

Product Rule

Goal: Understand and apply product rule.

Before you begin: Fractions, factorial notation and basic algebra.

Understand the idea

A sequence of choices can be represented by a branching tree. When every partial choice permits the same number of next options, multiplying those option counts gives the leaves.

total=m1 m2\displaystyle m_{1}\, m_{2}…mk\displaystyle m_{k}

Choose and carry out a method

Identify separate choice positions and count the available options at each. Multiply for choices that must all occur; add for disjoint alternative cases.

Check the reasoning

Check whether an earlier choice changes later options. Independence of numerical counts must be justified by the situation.

WORKED EXAMPLE 1

A code has three labeled slots. Choose one of 3 distinct letter labels for the first, one of 4 distinct number labels for the second, and one of 3 symbols for the third. How many codes are possible?

  1. Successive independent choices multiply.
  2. 3 choices can each be followed by 4 choices and then 3 choices.
  3. The total is 36. Count complete outcomes, not the sum of individual menus.

36

WORKED EXAMPLE 2

A code has three labeled slots. Choose one of 4 distinct letter labels for the first, one of 5 distinct number labels for the second, and one of 3 symbols for the third. How many codes are possible?

  1. Successive independent choices multiply.
  2. 4 choices can each be followed by 5 choices and then 3 choices.
  3. The total is 60. Count complete outcomes, not the sum of individual menus.

60

Common pitfalls

Possible mix-up: Add the option counts for a complete outfit.

A complete selection uses one option from each category, requiring multiplication.

Possible mix-up: A correct numerical answer alone explains the method.

State the governing relationship and check the conditions described above.

Explain it to yourself

Draw a small choice tree that explains a product.

Preview the eight practice prompts
  1. Problem 1

    A code has three labeled slots. Choose one of 6 distinct letter labels for the first, one of 7 distinct number labels for the second, and one of 3 symbols for the third. How many codes are possible?

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  2. Problem 2

    A code has three labeled slots. Choose one of 7 distinct letter labels for the first, one of 8 distinct number labels for the second, and one of 3 symbols for the third. How many codes are possible?

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  3. Problem 3

    A code has three labeled slots. Choose one of 8 distinct letter labels for the first, one of 9 distinct number labels for the second, and one of 3 symbols for the third. How many codes are possible?

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  4. Problem 4

    A code has three labeled slots. Choose one of 9 distinct letter labels for the first, one of 10 distinct number labels for the second, and one of 3 symbols for the third. How many codes are possible?

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  5. Problem 5

    A code has three labeled slots. Choose one of 10 distinct letter labels for the first, one of 11 distinct number labels for the second, and one of 3 symbols for the third. How many codes are possible?

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  6. Problem 6

    A code has three labeled slots. Choose one of 11 distinct letter labels for the first, one of 12 distinct number labels for the second, and one of 3 symbols for the third. How many codes are possible?

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  7. Problem 7New context

    A lunch allows one of 12 mains, one of 13 sides and one of 3 drinks. Every combination is available. How many different lunches can be ordered?

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  8. Problem 8New context

    A lunch allows one of 13 mains, one of 14 sides and one of 3 drinks. Every combination is available. How many different lunches can be ordered?

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Open stage CP 1.1 in the student workspace →

STAGE CP 1.2 · 8 PRACTICE PROBLEMS · 2 NEW-CONTEXT APPLICATIONS

Ordered Selections

Useful preparation: Product Rule

Goal: Understand and apply ordered selections.

Before you begin: Product Rule

Understand the idea

Filling distinct positions without replacement makes order significant. Each choice removes an available object, so the option counts decrease from one position to the next.

P (n,r)=n (n−1)\displaystyle P\, \left(n, r\right) = n\, \left(n - 1\right)…(n−r+1)\displaystyle \left(n - r + 1\right)

Choose and carry out a method

Fill the positions in order: n choices, then n−1\displaystyle n - 1, then n−2\displaystyle n - 2. Stop after the requested number of positions rather than always using all n objects.

Check the reasoning

Swapping two selected objects creates a different arrangement if their positions have distinct roles.

WORKED EXAMPLE 1

How many ordered lists of 3 different numbers can be selected from 1 through 5?

  1. Order matters and a chosen object cannot be selected again.
  2. The positions have 5, 4, and 3 choices.
  3. The total is 60. Different orders of the same three objects are different outcomes.

60

WORKED EXAMPLE 2

How many ordered lists of 3 different numbers can be selected from 1 through 6?

  1. Order matters and a chosen object cannot be selected again.
  2. The positions have 6, 5, and 4 choices.
  3. The total is 120. Different orders of the same three objects are different outcomes.

120

Common pitfalls

Possible mix-up: Use nr\displaystyle n^{r} without replacement.

The number of available choices decreases after each selection.

Possible mix-up: A correct numerical answer alone explains the method.

State the governing relationship and check the conditions described above.

Explain it to yourself

Compare selecting a team with assigning a captain, deputy and recorder.

Preview the eight practice prompts
  1. Problem 1

    How many ordered lists of 3 different numbers can be selected from 1 through 8?

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  2. Problem 2

    How many ordered lists of 3 different numbers can be selected from 1 through 9?

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  3. Problem 3

    How many ordered lists of 3 different numbers can be selected from 1 through 10?

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  4. Problem 4

    How many ordered lists of 3 different numbers can be selected from 1 through 11?

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  5. Problem 5

    How many ordered lists of 3 different numbers can be selected from 1 through 12?

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  6. Problem 6

    How many ordered lists of 3 different numbers can be selected from 1 through 13?

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  7. Problem 7New context

    14 runners race with no ties. How many possible ordered gold, silver and bronze podiums are there?

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  8. Problem 8New context

    15 runners race with no ties. How many possible ordered gold, silver and bronze podiums are there?

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Open stage CP 1.2 in the student workspace →

STAGE CP 1.3 · 8 PRACTICE PROBLEMS · 2 NEW-CONTEXT APPLICATIONS

Arrangements with Repetition

Useful preparation: Ordered Selections

Goal: Understand and apply arrangements with repetition.

Before you begin: Ordered Selections

Understand the idea

Labeling identical symbols temporarily overcounts each visible string. Permuting labels within a group of identical symbols does not change the visible arrangement.

(a+b)!a! b!=C (a+b,a)\displaystyle \frac{\left(a + b\right)!}{a!\, b!} = C\, \left(a + b, a\right)

Choose and carry out a method

Choose the positions occupied by one symbol, or divide the labeled permutation count by the factorials of the repeated group sizes.

Check the reasoning

Count a short example explicitly. Every visible arrangement must have the same number of labeled versions for the division argument to work.

WORKED EXAMPLE 1

How many distinct strings contain exactly 3 letters A and 2 letters B?

  1. Choose positions for one type; swapping identical objects does not make a new outcome.
  2. Choose 3 of the 5 positions: 5!3!⋅2!\displaystyle \frac{5!}{3! \cdot 2!}.
  3. There are 10 strings. The remaining positions automatically hold the other type.

10

WORKED EXAMPLE 2

How many distinct strings contain exactly 4 letters A and 2 letters B?

  1. Choose positions for one type; swapping identical objects does not make a new outcome.
  2. Choose 4 of the 6 positions: 6!4!⋅2!\displaystyle \frac{6!}{4! \cdot 2!}.
  3. There are 15 strings. The remaining positions automatically hold the other type.

15

Common pitfalls

Possible mix-up: Use (a+b)!\displaystyle \left(a + b\right)! for repeated symbols.

That treats indistinguishable copies as distinct.

Possible mix-up: A correct numerical answer alone explains the method.

State the governing relationship and check the conditions described above.

Explain it to yourself

Explain the same count by selecting positions for one symbol.

Preview the eight practice prompts
  1. Problem 1

    How many distinct strings contain exactly 2 letters A and 3 letters B?

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  2. Problem 2

    How many distinct strings contain exactly 3 letters A and 3 letters B?

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  3. Problem 3

    How many distinct strings contain exactly 4 letters A and 3 letters B?

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  4. Problem 4

    How many distinct strings contain exactly 5 letters A and 3 letters B?

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  5. Problem 5

    How many distinct strings contain exactly 2 letters A and 4 letters B?

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  6. Problem 6

    How many distinct strings contain exactly 3 letters A and 4 letters B?

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  7. Problem 7New context

    A row of 8 signal lights contains 4 red and 4 blue lights. Lights of the same color are identical. How many color patterns can it show?

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  8. Problem 8New context

    A row of 9 signal lights contains 5 red and 4 blue lights. Lights of the same color are identical. How many color patterns can it show?

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Open stage CP 1.3 in the student workspace →
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