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Count Cookie Flavor Combinations

SIG (Susquehanna)Data Scientist · Online Assessment
EasyData ScientistOnline AssessmentStatistics & Math

Last updated

Calculate the number of distinct unordered combinations when selecting 5 cookies from 4 available flavors with replacement, and contrast this with the ordered outcome space.

Problem Description

A bakery produces 4 distinct flavors of cookies: Chocolate Chip, Oatmeal Raisin, Snickerdoodle, and Peanut Butter. An unlimited supply of each flavor is available.

You want to purchase a box containing a total of 5 cookies. Only the final count of each flavor in the box matters—the arrangement or order in which the cookies are placed into the box is irrelevant.

Tasks

  1. Combination Count: Compute the total number of distinct boxes of 5 cookies that can be formed using the 4 flavors.
  2. Algebraic & Combinatorial Formulation: Formulate this problem as the number of non-negative integer solutions to an algebraic equation. State the formula via the "stars and bars" method and show the step-by-step arithmetic.
  3. Ordered vs. Unordered Contrast: Suppose the box has 5 distinct, labeled compartments (order matters). Compute the total number of distinct arrangements possible, and explain why this differs significantly from the unordered total.

Hints

Hint 1

Express the problem as finding the number of non-negative integer solutions to x_1 + x_2 + x_3 + x_4 = 5, where each variable represents the count of a flavor.

Hint 2

Use the stars and bars formula: the number of ways to distribute n identical items into k distinct bins is C(n + k - 1, k - 1).

Hint 3

For the ordered variation, each of the 5 independent slots can be filled by any of the 4 flavors without dependency.

Follow-up questions

  • How does the number of valid combinations change if the box must include at least one cookie of each flavor?
  • If you select 5 cookies sequentially with replacement where each flavor is equally likely at each draw, does every unordered combination have the same probability of occurring?
  • How would you compute the count if there were an upper bound constraint, such as at most 2 Peanut Butter cookies allowed?