Permutations and Combinations

Choosing a shirt (3 options) and pants (4 options): 3 × 4 = 12 ways, Rolling two dice: 6 × 6 = 36 outcomes, Creating a password with 2 letters then 3 digits: 26 × 26 × 10 × 10 × 10

The multiplication principle applies when events occur in succession. Option 1: Correct - shirt AND pants. Option 2: Correct - first die AND second die. Option 3: Incorrect - this is OR (addition principle). Option 4: Correct - each position filled successively. Option 5: Incorrect - this is OR (addition principle).

125

Step 1: MANGO has 5 distinct letters: M, A, N, G, O. Step 2: Since repetition is allowed, each of the 3 positions can be filled with any of the 5 letters. Step 3: Total words = 5 × 5 × 5 = 5³ = 125. Common mistake: Without repetition would be ⁵P₃ = 60.

8

Step 1: Use the property ⁿCᵣ = ⁿCₙ₋ᵣ. Step 2: If ⁿC₅ = ⁿC₃, then either 5 = 3 (impossible) or 5 = n - 3. Step 3: From 5 = n - 3, we get n = 8. Step 4: Verify: ⁸C₅ = ⁸C₃ = 56 ✓

120

Step 1: For circular arrangements, we fix one person's position to account for rotational symmetry. Step 2: Remaining (n-1) people can be arranged in (n-1)! ways. Step 3: For 6 people: (6-1)! = 5! = 120. Common mistake: 6! = 720 (this is for linear arrangement).