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Fashion Brand Photo Shoot

📸 Campaign Lineup Planner A combinatorics problem: Permutations with adjacency restrictions. Director's Rules 1 Models wearing Headwear (Caps or Beanies) CANNOT stand next to each other. Headwear group: Bella, Carlos, Ethan, Hannah. 2 Alex (Tall) and Fiona (Short) CANNOT stand next to each other. Sample Valid Lineup Hannah Beanie Daisy Bella Cap Gabe Carlos Beanie Fiona (Short) Ethan Cap Alex (Tall) If the photographer randomly arranges the 8 models in a line, what is the PROBABILITY that the lineup meets all conditions?

A fashion brand is organizing a photo shoot for their new advertising campaign. They have eight models: Alex, Bella, Carlos, Daisy, Ethan, Fiona, Gabe, and Hannah. The brand wants to showcase their full outfits, including hair accessories like caps and beanies. However, due to aesthetic considerations, they have some restrictions:

1. Models wearing headwear (caps or beanies) cannot stand next to each other in the photo.
2. Alex (who is very tall) and Fiona (who is quite short) cannot stand next to each other.

The following models are wearing headwear: Bella (cap), Carlos (beanie), Ethan (cap), and Hannah (beanie).

If the photographer randomly arranges the models in a line, what is the probability that the arrangement will meet all the required conditions?

The diagram above illustrates one possible valid arrangement of the models.

Solution

To solve this probability problem, we’ll follow a systematic approach involving combinatorics.

**Given:**
– Models with headwear: Bella (B), Carlos (C), Ethan (E), Hannah (H)
– Models without headwear: Alex (A), Daisy (D), Fiona (F), Gabe (G)
– Constraints:
1. Models wearing headwear cannot stand next to each other.
2. Alex (A) and Fiona (F) cannot stand next to each other.

**Step 1: Total Number of Arrangements Without Restrictions**
Total arrangements = 8! = 40,320

**Step 2: Arranging Non-Headwear Models (A, D, F, G) Without A and F Adjacent**
– Total ways without restrictions: 4! = 24
– Total ways with A and F adjacent: 3! × 2 = 12 (treating A-F as a unit, then multiplying by 2 for A-F and F-A)
– Valid arrangements: 24 – 12 = 12

**Step 3: Placing Headwear Models in the Gaps**
– Number of gaps: 5 (before, between, and after non-headwear models)
– Ways to choose 4 gaps out of 5: C(5,4) = 5
– Ways to arrange 4 headwear models: 4! = 24
– Total ways to place headwear models: 5 × 24 = 120

**Step 4: Calculating Total Valid Arrangements**
Total valid arrangements = 12 × 120 = 1,440

**Step 5: Computing the Probability**
Probability = 1,440 / 40,320 = 1/28 ≈ 0.0357 or about 3.57%

**Answer:** The probability that the photographer will randomly arrange the models in a valid order is 1/28 or approximately 3.57%.

✅ Solution: Combinatorics Breakdown Using the Complement Method and the Gap Method to find valid permutations. 1️⃣ Arrange Non-Headwear Models (A, D, F, G) We use the Complement Method: Total arrangements minus the invalid ones (Alex & Fiona touching). Total (No Rules) 4! = 24 - Invalid (A & F Together) 3! × 2 = 12 = Valid Base 12 Ways 2️⃣ Place Headwear Models in the "Gaps" To ensure headwear models (B, C, E, H) never touch, insert them into the 5 available gaps. 1 2 3 4 5 Choose 4 Gaps: C(5,4) = 5 × Arrange 4 Models: 4! = 24 = 120 Ways 3️⃣ Final Probability Calculation Total Valid Arrangements: 12 (Base) × 120 (Gaps) = 1,440 Probability (Valid / Total): 1,440 ÷ 40,320 (8!) = 1 / 28 PROBABILITY OF A VALID LINEUP 1/28 (≈ 3.57%)