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Difficulty: MediumFundamental Counting Principle

An artisan tea shop allows customers to order a custom tea blend prepared through three sequential choices:

1. Base Tea: Choose 11 of 44 available tea types (Black, Green, Oolong, or White).
2. Flavor Infusion: Choose 11 of 55 available herbal flavors. However, if Black tea is chosen as the base, only 33 of these 55 herbal flavors can be selected.
3. Sweetener: Choose 11 of 33 natural sweeteners (Honey, Stevia, or Maple Syrup), or choose to have no sweetener.

How many different custom tea blends can a customer create?

  1. A
    5454
  2. B
    6060
  3. 7272Answer
  4. D
    8080
  5. E
    9696

Answer

72 different custom tea blends can be created.
To solve this counting problem with conditional restrictions, break the calculation into two mutually exclusive cases based on the base tea chosen:

1. Case 1: Black Tea Base
- Base tea options: 11 (Black)
- Flavor infusion options: 33 (restricted subset)
- Sweetener options: 44 (33 sweeteners + 11 option for no sweetener)
- Combinations = 1×3×4=121 \times 3 \times 4 = 12

2. Case 2: Non-Black Tea Base (Green, Oolong, or White)
- Base tea options: 33
- Flavor infusion options: 55 (all available)
- Sweetener options: 44
- Combinations = 3×5×4=603 \times 5 \times 4 = 60

Adding the combinations from these two mutually exclusive cases yields 12+60=7212 + 60 = 72 total custom tea blends.

Step-by-Step Solution

1
Identify total options available for the sweetener stage.
There are 33 sweetener choices plus 11 choice for 'no sweetener', making 3+1=43 + 1 = 4 total options for Stage 3.
Choosing 'no sweetener' is a distinct decision outcome that must be counted.
2
Calculate the number of blends when Black tea is chosen as the base.
Number of Black tea blends = 1 (Black base)×3 (flavors)×4 (sweetener choices)=121 \text{ (Black base)} \times 3 \text{ (flavors)} \times 4 \text{ (sweetener choices)} = 12.
When Black tea is chosen, only 33 flavor infusions are permitted.
3
Calculate the number of blends when any non-Black tea is chosen as the base.
Number of non-Black tea blends = 3 (Green, Oolong, White bases)×5 (flavors)×4 (sweetener choices)=603 \text{ (Green, Oolong, White bases)} \times 5 \text{ (flavors)} \times 4 \text{ (sweetener choices)} = 60.
There are 33 non-Black base choices, and each can be paired with any of the 55 flavor infusions.
4
Sum the combinations from both mutually exclusive cases.
Total blends = 12+60=7212 + 60 = 72.
Since selecting Black tea and selecting non-Black tea are mutually exclusive cases, their individual counts are added.

Key Concept

Fundamental Counting Principle with Conditional Restrictions
Estimated Time:1m 45s
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Topics

Fundamental Counting Principle

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