At a certain restaurant, each sandwich served has three components-bread, topping, and filling. A customer orders a sandwich by choosing...

Phase 1: Owning the Dataset

Visual Representation

Sandwich Components:

Bread Options: [1] [2] [3] [4] [5] [6] [7] [8] = 8 options
Topping Options: [?] = T options (T < F)
Filling Options: [?] = F options

Total Combinations = \(8 \times \mathrm{T} \times \mathrm{F} = 1536\)

Key Given Information:

• Each sandwich needs exactly one of each component
• 8 bread options
• Fewer topping options than filling options (\(\mathrm{T} < \mathrm{F}\))
• Total possible sandwiches = 1,536

Phase 2: Understanding the Question

Setting Up the Equation

Since total combinations = bread × topping × filling:
\(8 \times \mathrm{T} \times \mathrm{F} = 1536\)

Dividing both sides by 8:
\(\mathrm{T} \times \mathrm{F} = 1536 ÷ 8 = 192\)

What We're Looking For:

• Two numbers from the choices that multiply to 192
• The smaller number = toppings
• The larger number = fillings

Phase 3: Finding the Answer

Systematic Check of Options

We need pairs from [6, 12, 24, 36, 48, 8] that multiply to 192.

Let's check systematically:

• \(6 \times ? = 192 → ? = 32\) (not in choices)
• \(8 \times ? = 192 → ? = 24\) ✓ (24 is in choices!)
• \(12 \times ? = 192 → ? = 16\) (not in choices)
• \(24 \times ? = 192 → ? = 8\) ✓ (8 is in choices!)

? Stop here - we found valid pairs: (8,24) and (24,8)

Applying the Constraint

Since toppings < fillings:

• If T = 8 and F = 24: \(8 < 24\) ✓
• If T = 24 and F = 8: \(24 < 8\) ✗

Therefore:

• Number of toppings = 8
• Number of fillings = 24

Verification

\(8 \text{ bread} \times 8 \text{ toppings} \times 24 \text{ fillings} = 8 \times 8 \times 24 = 1536\) ✓

Phase 4: Solution

Statement 1 (Number of toppings): 8
Statement 2 (Number of fillings): 24

These values satisfy all conditions:

• Their product equals 192
• Toppings (8) < Fillings (24)
• Total combinations = \(8 \times 8 \times 24 = 1536\)