How Much to Pack?

This puzzle features combinatorics to decide a finite set.

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.

Challenge 1

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.

Challenge 2

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?

Background

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 such as 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.

Course:

Math [1]

Result/Solution(s)

Challenge 1

Sarina needs 14 different outfits, since she plans to change outfits once each day. So she needs at least two tops and two bottoms. Let’s call these T1, T2, and B1, B2.

How_Much_to_Pack_1

Construct a table listing all the possible outfits she can wear with two tops and two bottoms.


	T1	T2
B1	T1-B1	T2-B1
B2	T1-B2	T2-B2

How_much_to_pack_2

Two tops and two bottoms provide only four possible outfits.

Let’s experiment by keeping the number of tops constant and see how many bottoms Sarina needs.

With two tops and three bottoms we get six outfits:


	T1	T2
B1	T1-B1	T2-B1
B2	T1-B2	T2-B2
B3	T1-B3	T2-B3

Following through with this pattern, we see that Sarina can have 14 different outfits if she takes seven bottoms to go with her two tops. So she has to pack nine garments in all.

How_much_to_pack_3


	T1	T2
B1	T1-B1	T2-B1
B2	T1-B2	T2-B2
B3	T1-B3	T2-B3
B4	T1-B4	T2-B4
B5	T1-B5	T2-B5
B6	T1-B6	T2-B6
B7	T1-B7	T2-B7

Sarina wants to pack the smallest number of total garments. Is there any way she can have 14 different outfits with a smaller total number of garments?

By this time you’ve probably figured out that the number of outfits is the product of the number of tops and bottoms. So if Sarina packs either three tops and five bottoms or five tops and three bottoms, she’ll have 15 different possible outfits. That’s a total of eight garments she has to pack.

Three tops and five bottoms

How_much_to_pack_5

How_much_to_pack_6

Five tops and three bottoms

How_much_to_pack_7

How_much_to_pack_8

There’s one more way Sarina can have enough outfits with eight garments: four tops and four bottoms make a total of 16 different outfits.

Four tops and four bottoms

How_much_to_pack_9

How_much_to_pack_10

So we have found three different ways that Sarina can pack eight garments so that she can vary her outfits twice a day and never wear the same outfit more than once in a week.

Challenge 2

For this challenge Sarina plans to change from pants to skirt, or vice versa, every day. The reasoning is similar, but with a different constraint, the results may change. For example, Sarina needs seven different skirt-blouse outfits and seven different pants-blouse outfits.

How_much_to_pack_11

As before, Sarina needs at least two tops so she can change tops once a day. To get seven outfits, she can pack four skirts and two tops for eight skirt-blouse outfits and four additional pants to make eight pants-blouse outfits. Four skirts, four pairs of pants, and two tops make ten garments all together.

How_much_to_pack_12

Can she do it with fewer garments overall?

Suppose Sarina packs three tops. Then she’ll need three skirts (nine skirt-blouse outfits) and three pairs of pants (nine pants-blouse outfits). This is a total of nine garments, in the exercise above. This is one less than with two tops and four of each kind of bottom. But can she do it with fewer than nine garments?

Nine garments

How_much_to_pack_13

How_much_to_pack_14

Suppose Sarina packs four tops. Then she needs only two skirts and two pairs of pants to have eight of each type of outfit. Four tops plus two skirts and two pants equal eight garments in all.

Eight garments

How_much_to_pack_15

How_much_to_pack_16

Therefore the smallest number of total garments is still eight! But we found three different ways to have eight garments for Challenge 1. There is only one way to do it for Challenge 2.

combination problems [2]