Twas Brillig has a drawer containing a mixture of 15 black...

Twas Brillig has a drawer containing a mixture of 15 black and 20 blue socks. Twas is sick in bed when his friend Slithy Toves comes to visit. (a) Twas asks Slithy to get him a pair of matched socks from the drawer (either a black pair or a blue pair). It is too dark for Slithy to distinguish the colors. How many socks must Slithy remove from the drawer to be sure of getting a matched pair? (b) Suppose now there are an equal number of black and blue socks in the drawer. Suppose the minimum number of socks Slithy must draw to be sure of getting a pair is the same as the minimum number he must draw to be sure of getting at least one black sock and one blue sock. How many socks are in the drawer?

Twas Brillig has a drawer containing a mixture of 15 black and 20 blue socks. Twas is sick in bed when his friend Slithy Toves comes to visit.
(a) Twas asks Slithy to get him a pair of matched socks from the drawer (either a black pair or a blue pair). It is too dark for Slithy to distinguish the colors. How many socks must Slithy remove from the drawer to be sure of getting a matched pair?
(b) Suppose now there are an equal number of black and blue socks in the drawer. Suppose the minimum number of socks Slithy must draw to be sure of getting a pair is the same as the minimum number he must draw to be sure of getting at least one black sock and one blue sock. How many socks are in the drawer?

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Key Concepts

Pigeonhole Principle

The pigeonhole principle is a fundamental concept in combinatorics which states that if items are distributed among a limited number of categories or 'pigeonholes', then at least one category must contain more than one item when the number of items exceeds the number of categories. In selection problems, this principle is used to guarantee a repeated element, such as drawing a matching pair from a collection of items grouped by color.

Worst-Case Scenario Analysis

Worst-case scenario analysis involves considering the most unfavorable sequence of events when making random selections. In problems that ask for a guarantee (such as ensuring a matching pair or specific diversity in choices), the analysis helps in determining the minimum number of selections required to overcome any possible adverse outcome in the draw.

Combinatorial Reasoning

Combinatorial reasoning refers to the systematic counting and arrangement techniques used to analyze problems involving discrete objects. In the context of selection problems, it aids in understanding how various outcomes can occur and in formulating conditions under which certain events, like obtaining matching pairs or specific combinations, are guaranteed.

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