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Counting Applications worksheets

Choose among the multiplication principle, ordered selections without replacement, and unordered combinations in varied finite contexts.

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Identify whether order matters and whether repetition is allowed. Name the counting method, show the factors or formula, and give the exact count.

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Connect counting decisions with probability models

Counting sheets make order and repetition policies explicit. Conditional-probability sheets keep the conditioning population visible in complete two-way tables and update both category counts and denominators after a draw without replacement.

Learning objective

What this activity builds

Identify whether order and repetition matter, then use the appropriate bounded counting method to find an exact total.

Worked example

From 7 students, how many different 3-person committees can be chosen?

7C3 = 35.

Committee order does not matter and no student repeats, so divide the 7P3 ordered selections by 3!.

Teaching notes

Make the practice count

  • Have students mark order matters or order does not matter before calculating.
  • State the repetition rule beside every multiplication-principle problem.
  • Check small cases by listing outcomes before relying on a formula.

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Math SheetNameScoreTeacherDateCounting ApplicationsIdentify whether order matters and whether repetition is allowed. Name the counting method, show the factors or formula, and give the exact count.1.From 6 different books, how many sets of 3 books can be selected? Identify the counting method.Selection policy: No person or item may be selected more than once.2.A trip has 3 first-leg routes, 6 second-leg routes, and 4 final-leg routes. How many complete routes are possible? Identify the countingmethod.Selection policy: Each stage is chosen independently; reusing a category choice does not remove later choices.Use exact counts and reduce every probability fraction.math-sheet.comPage 1 of 4
Math SheetNameScoreTeacherDateCounting ApplicationsIdentify whether order matters and whether repetition is allowed. Name the counting method, show the factors or formula, and give the exact count.3.A club with 12 members fills 3 distinct officer roles. How many assignments are possible? Identify the counting method.Selection policy: No person or item may be selected more than once.4.A lunch menu has 5 entrees, 4 sides, and 2 drinks. How many meals choose one of each? Identify the counting method.Selection policy: Each stage is chosen independently; reusing a category choice does not remove later choices.Use exact counts and reduce every probability fraction.math-sheet.comPage 2 of 4
Math SheetNameScoreTeacherDateCounting ApplicationsIdentify whether order matters and whether repetition is allowed. Name the counting method, show the factors or formula, and give the exact count.5.From 6 distinct toppings, how many selections of 5 different toppings are possible? Identify the counting method.Selection policy: No person or item may be selected more than once.6.From 10 speakers, how many ordered schedules of 5 different speakers are possible? Identify the counting method.Selection policy: No person or item may be selected more than once.Use exact counts and reduce every probability fraction.math-sheet.comPage 3 of 4
Math SheetNameScoreTeacherDateCounting ApplicationsIdentify whether order matters and whether repetition is allowed. Name the counting method, show the factors or formula, and give the exact count.7.A three-position code uses 5, 4, and 3 allowed symbols by position. Symbols may repeat. How many codes are possible? Identify thecounting method.Selection policy: Repetition is allowed, and each position has its stated set of choices.8.From 12 volunteers, how many different committees of 2 people are possible? Identify the counting method.Selection policy: No person or item may be selected more than once.Use exact counts and reduce every probability fraction.math-sheet.comPage 4 of 4

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