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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 to the right illustrates one possible valid arrangement of the models.
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%.
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