Counting Methods Practice Worksheet
Finite Math — Lesson 4-1 Worksheet Assignment
Fundamental Counting Principle, Permutations & Repeated Arrangements
Name:
Date: Period:
Score: ____________
FCP (Single Item)
\(N = n_1 \cdot n_2 \dots n_k\)
Permutation (Order Matters)
\(_{n}P_{r} = \frac{n!}{(n-r)!}\)
Combination (No Order)
\(_{n}C_{r} = \frac{n!}{r!(n-r)!}\)
Repeated Elements
\(\frac{n!}{n_1! \, n_2! \dots n_k!}\)
1
Part 1: Fundamental Counting Principle
Selecting 1 item per category
For questions 1–4, identify the number of possibilities for each independent stage and calculate the total number of distinct outcomes.
1. Hoosier Deli Boxed Lunch FCP
A customer orders a custom boxed lunch by choosing exactly 1 bread (sourdough, wheat, rye, brioche), 1 deli meat (turkey, ham, roast beef), 1 cheese (cheddar, swiss, provolone, pepper jack), 1 side (chips, apple, pasta salad), and 1 drink (tea, lemonade, soda, water).
__ × __ × __ × __ × __ =
Total:
2. Indiana High School Student ID FCP
A school issues a 5-character ID card. The first 2 characters are uppercase English letters with no letters repeated. The final 3 characters are digits from 0 through 9 where digits may repeat freely.
__ × __ × __ × __ × __ =
Total:
3. Custom Laptop Configuration FCP
A student builds a laptop by choosing 1 processor (i5, i7, i9), 1 RAM size (8GB, 16GB, 32GB, 64GB), 1 solid-state drive (512GB, 1TB, 2TB), 1 finish (silver, space gray, matte black), and 1 warranty tier (1-yr, 2-yr, 3-yr).
__ × __ × __ × __ × __ =
Total:
4. ATM Security Access PIN FCP
A bank customer must set a 4-digit personal identification number (PIN) using the digits 0 through 9. Due to bank security protocol, the first digit cannot be 0 and no digit may be repeated.
__ × __ × __ × __ =
Total:
2
Part 2: Permutations with Repeated Elements
Multinomial arrangements
For questions 5–6, find the number of distinct letter arrangements. Show your formula setup with factorials and final calculation.
5. Word: "INDIANA" 7 Letters Total
Work & Factorial Setup:
Distinct Arrangements
Answer:
6. Word: "BASKETBALL" 10 Letters Total
Work & Factorial Setup:
Distinct Arrangements
Answer:
Finite Math • Lesson 4-1 Worksheet Assignment Turn to Page 2 for Permutations vs. Combinations → Page 1 of 3
3
PART 3: PERMUTATION OR COMBINATION?
Randomized Mix • 8 Problems
For each scenario below: (1) Circle whether it is a Permutation [ P ] or Combination [ C ], (2) Check whether order matters, (3) Write standard notation (\(_{n}P_{r}\) or \(_{n}C_{r}\)), and (4) Compute the total possible outcomes.
7. Math Club Officer Election
P C
A high school finite math club has 15 members. In how many ways can the club elect a President, Vice President, Secretary, and Treasurer?
Order: [ ] Matters [ ] Does Not Matter \(n = \_\_\_\), \(r = \_\_\_\)
Setup: \(_{n}\boxed{\phantom{X}}_{r} = \)
Outcomes:
8. Academic Decathlon Team
P C
From a pool of 24 qualified students, the faculty advisor must select a team of 5 students to represent the school at the state competition.
Order: [ ] Matters [ ] Does Not Matter \(n = \_\_\_\), \(r = \_\_\_\)
Setup: \(_{n}\boxed{\phantom{X}}_{r} = \)
Outcomes:
9. Track 4×400m Relay Lineup
P C
A track coach has 9 sprinters. In how many ways can she choose 4 sprinters and assign their exact running positions (Lead-off, 2nd leg, 3rd leg, Anchor)?
Order: [ ] Matters [ ] Does Not Matter \(n = \_\_\_\), \(r = \_\_\_\)
Setup: \(_{n}\boxed{\phantom{X}}_{r} = \)
Outcomes:
10. Pizza Special Topping Selection
P C
A local pizzeria offers a 4-topping deal. If there are 14 distinct toppings available on the menu, how many different 4-topping pizzas can be ordered?
Order: [ ] Matters [ ] Does Not Matter \(n = \_\_\_\), \(r = \_\_\_\)
Setup: \(_{n}\boxed{\phantom{X}}_{r} = \)
Outcomes:
11. Science Fair Ribbon Awards
P C
At the regional science symposium, judges evaluate 18 finalists. In how many different ways can the 1st Place, 2nd Place, and 3rd Place ribbons be awarded?
Order: [ ] Matters [ ] Does Not Matter \(n = \_\_\_\), \(r = \_\_\_\)
Setup: \(_{n}\boxed{\phantom{X}}_{r} = \)
Outcomes:
12. Art Gallery Wall Hanging
P C
A curator has 11 paintings submitted for an exhibition and must choose 4 of them to hang in a straight horizontal display row from left to right.
Order: [ ] Matters [ ] Does Not Matter \(n = \_\_\_\), \(r = \_\_\_\)
Setup: \(_{n}\boxed{\phantom{X}}_{r} = \)
Outcomes:
13. AP Seminar Reading List
P C
An English teacher provides a master list of 16 classic novels. A student must select any 4 novels to read and analyze over the semester.
Order: [ ] Matters [ ] Does Not Matter \(n = \_\_\_\), \(r = \_\_\_\)
Setup: \(_{n}\boxed{\phantom{X}}_{r} = \)
Outcomes:
14. Jury Pool Selection
P C
From a courtroom venire of 20 prospective jurors, a judge and attorneys empanel a final group of 6 jurors with no internal ranks or roles.
Order: [ ] Matters [ ] Does Not Matter \(n = \_\_\_\), \(r = \_\_\_\)
Setup: \(_{n}\boxed{\phantom{X}}_{r} = \)
Outcomes:
Finite Mathematics • Permutations vs. Combinations Turn to Page 3 for Multi-Stage Compound Counting → Page 2 of 3
4
PART 4: MULTI-STAGE COMPOUND COUNTING
FCP & Combinations
In these complex multi-decision problems, you must select multiple items within at least one category, set up the appropriate counting methods yourself, and then apply the Fundamental Counting Principle to find total possibilities.
15. The Bloomington Artisan Ice Cream Flight Compound Decision
A customer at a gourmet creamery designs a custom sampler sundae. The order requires four consecutive decisions:
- Stage 1 (Vessel): Choose 1 cone/bowl style out of 4 options.
- Stage 2 (Flavors): Choose 3 different gelato flavors from a menu of 12 flavors (order does not matter).
- Stage 3 (Sauces): Choose 2 different hot dessert sauces from 8 gourmet sauces (order does not matter).
- Stage 4 (Garnish): Choose 1 signature topping sprinkle from 5 choices.
Stage 1: Vessel
Select 1 of 4
Ways: ______
Stage 2: Flavors
Select 3 of 12
Ways: ______
Stage 3: Sauces
Select 2 of 8
Ways: ______
Stage 4: Garnish
Select 1 of 5
Ways: ______
FCP Calculation: (___) × (___) × (___) × (___) =
Total Sundae Possibilities:
16. Eli Lilly Biomedical Launch Task Force Compound Decision
An Indianapolis biotechnology company is assembling a 6-person clinical launch committee from three employee groups: 10 Biochemists, 8 Biostatisticians, and 6 Regulatory Specialists. The committee must consist of:
- Exactly 2 Biochemists chosen from the 10 available biochemists.
- Exactly 2 Biostatisticians chosen from the 8 available biostatisticians.
- Exactly 2 Regulatory Specialists chosen from the 6 available regulatory specialists.
Group 1: Biochemists
Choose 2 from 10
Ways: ______
Group 2: Biostatisticians
Choose 2 from 8
Ways: ______
Group 3: Regulatory Specialists
Choose 2 from 6
Ways: ______
FCP Calculation: (___) × (___) × (___) =
Total Committee Combinations:
17. Finite Math Synthesis Challenge: High-Security Cryptographic Token Synthesis Challenge
A software security key contains three distinct verification segments generated simultaneously:
- Segment A: Select 1 security symbol from 6 special characters: {@, #, $, %, &, *}.
- Segment B: Select 3 distinct letters from the 26 uppercase English letters where sequence / order matters.
- Segment C: Select 2 distinct odd digits from {1, 3, 5, 7, 9} where order does NOT matter.
Segment B Type: [ ] Permutation [ ] Combination • Setup: \(_{26}\boxed{\phantom{X}}_{3} = \_\_\_\_\_\_\)
Segment C Type: [ ] Permutation [ ] Combination • Setup: \(_{5}\boxed{\phantom{X}}_{2} = \_\_\_\_\_\_\)
Total Valid Keys:
Finite Math • Lesson 4-1 Worksheet Assignment End of Assignment • Verify all calculations Page 3 of 3
Counting Methods Answer Key
Teacher Solutions Manual Finite Math • Lesson 4-1 Answer Key
Finite Math — Lesson 4-1 Answer Key
Fundamental Counting Principle, Permutations & Repeated Arrangements
Assignment: Lesson 4-1
Problems: 17 Total
Format: Step-by-Step
FCP Rule
\(N = n_1 \cdot n_2 \dots n_k\)
Permutation Formula
\(_{n}P_{r} = \frac{n!}{(n-r)!}\)
Combination Formula
\(_{n}C_{r} = \frac{n!}{r!(n-r)!}\)
Repeated Elements
\(\frac{n!}{n_1! \, n_2! \dots n_k!}\)
1
Part 1 Solutions: Fundamental Counting Principle
Single Selection per Stage
1. Hoosier Deli Boxed Lunch Solved
Bread (4) • Meat (3) • Cheese (4) • Side (3) • Drink (4)
\(4 \times 3 \times 4 \times 3 \times 4 = 12 \times 12 \times 4\)
Total Boxed Lunches
Answer:
576
2. Indiana High School Student ID Solved
Letter 1 (26) • Letter 2 no repeat (25) • Digits with repeat (10 × 10 × 10)
\(26 \times 25 \times 10 \times 10 \times 10 = 650 \times 1{,}000\)
Total Unique IDs
Answer:
650,000
3. Custom Laptop Configuration Solved
CPU (3) • RAM (4) • SSD (3) • Finish (3) • Warranty (3)
\(3 \times 4 \times 3 \times 3 \times 3 = 12 \times 27\)
Total Configurations
Answer:
324
4. ATM Security Access PIN Solved
Digit 1 no zero (9) • Digit 2 (9) • Digit 3 (8) • Digit 4 (7)
\(9 \times 9 \times 8 \times 7 = 81 \times 56\)
Total Valid PINs
Answer:
4,536
2
Part 2 Solutions: Permutations with Repeated Letters
Multinomial Expansion
5. Word: "INDIANA" \(n = 7\) Total
Letter Counts: I: 2, N: 2, D: 1, A: 2
\(\frac{7!}{2! \, 2! \, 1! \, 2!} = \frac{5{,}040}{2 \cdot 2 \cdot 1 \cdot 2} = \frac{5{,}040}{8}\)
Unique Arrangements
Answer:
630
6. Word: "BASKETBALL" \(n = 10\) Total
Letter Counts: B: 2, A: 2, L: 2 (S, K, E, T = 1)
\(\frac{10!}{2! \, 2! \, 2! \, 1! \dots} = \frac{3{,}628{,}800}{2 \cdot 2 \cdot 2} = \frac{3{,}628{,}800}{8}\)
Unique Arrangements
Answer:
453,600
Teacher Solutions • Finite Math Lesson 4-1 Turn to Page 2 for Permutations vs. Combinations → Key • Page 1 of 3
