Sarina is packing her clothes for a week’s vacation with her family. She plans to wear different outfits every day. An outfit is a combination of a blouse and either pants or a skirt. Also, Sarina likes to dress differently for the evening than during the day. She does not want to wear the same top or bottom twice in one day.
What is the smallest number of garments—blouses, pants, and/or skirts—that Sarina needs to pack so that she can wear two different outfits each day, and never wear the same outfit twice? Any particular blouse, pants, or skirt can be worn more than once as long as it is combined into a different outfit.
Sarina wants to wear pants in the daytime and switch to a skirt at night, or vice versa, to wear a skirt during the day and change to pants at night. How many skirts and pants does she need? Does this change the number of garments she needs to pack?
Combinatorics is a “branch of mathematics concerned with the selection, arrangement, and combination of objects chosen from a finite set.” (Britannica Concise Encyclopedia) The number of different ways to deal a hand in a card game like poker, bridge, or solitaire is a simple example (although it’s a big number). Assigning students and scheduling classes is another example. There are few standard algorithms for problems in combinatorics. Instead, each problem requires its own logical analysis—this makes combinatorics a great field for finding interesting math puzzles. The development of computer networks with so many different codes and passwords makes combinatorics one of the most important mathematical fields of our day.
Last month’s problem, How Many License Plates, involved combinations of numbers and letters. Although this month’s problem has fewer possibilities it also requires careful analysis and consideration of all cases to come up with a solution.
After you’ve tried this for yourself check our solutions. |
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