Each exercise includes the core concept, the required formula, the inputs, and space to try. Click "Show Answer" to see the solution after attempting!
1. Concept: Permutation (Order Matters) Formula:P(n, r) = n! / (n - r)! Inputs: n = 6, r = 2 Question: How many ways can you arrange 2 out of 6 books in a row?
Answer: P(6, 2) = 6 × 5 = 30 Excel:=PERMUT(6, 2)
2. Concept: Combination (Order Doesn’t Matter) Formula:C(n, r) = n! / [r!(n - r)!] Inputs: n = 7, r = 3 Question: How many ways can you choose 3 team members from 7?
Answer: C(7, 3) = 35 Excel:=COMBIN(7, 3)
3. Concept: Multiplication Rule Formula:n₁ × n₂ × ... × nₖ Inputs: 3 shirts × 2 pants × 4 shoes Question: How many different outfits can be made?
Answer: 3 × 2 × 4 = 24 Excel:=3*2*4
4. Concept: Combination (choose positions) Formula:C(n, r) = n! / (r!(n - r)!) Inputs: 5 surgeries: 3 knees (k), 2 hips (h) Question: How many surgery orders are possible?
Answer: Choose which 3 of the 5 slots are knees (the rest are hips): C(5, 3) = 10 Excel:=COMBIN(5, 3)
Note: Equivalent to =FACT(5)/(FACT(3)*FACT(2)) (the two-type identical-items formula).
5. Concept: Combination – Subsets Formula:C(n, r) Inputs: n = 8, r = 5 Question: How many ways to place 5 identical chips in 8 slots?
Answer: C(8, 5) = 56 Excel:=COMBIN(8, 5)
6. Concept: Sampling Without Replacement Formula:C(def, k) × C(non-def, rest) Inputs: 3 defective, 47 good, choose 6, exactly 2 defective Question: How many such samples?
7. Concept: Permutation (Subset, Order Matters) Formula:P(9, 3) = 9 × 8 × 7 Inputs: n = 9, r = 3 Question: How many ways to assign gold, silver, bronze medals to 3 out of 9 athletes?
Answer: P(9, 3) = 504 Excel:=PERMUT(9, 3)
8. Concept: Combination – No Order Formula:C(10, 2) = 10! / (2! × 8!) Inputs: n = 10, r = 2 Question: How many unique handshakes between 10 people?