"Give them the gift of words"

Jun
1st

Fun Math Problems With Permutations & Combinations

Categories: Math |


Problems related to permutations and combinations are often more fun to solve. There are generally no equations, no diagrams, no graphs. You just have to translate the words into the situation at hand. Words like ‘at least’ and ‘at most’ make a huge difference, as you know. For example,

Q: 10 students in a class of 60 students study French, while 47 students study German.What is the minimum/maximum number of students who study Japanese if at the most/ at least 15 students study more than one language and everybody studies at least one language?

As you can see, there is a variety of answers possible, based on the choice of words. But this was a fairly straightforward question that you can solve by plugging in answer values, or actually deriving the answer. It was also easy to translate the words into relations.

A slightly harder question would require some extraneous knowledge, maybe knowledge of geometry or algebra.

Q: What is the number of possible regular geometric figures with integer sides and an area of 16 units? Assume that the number of sides is less than 7.

A regular geometric figure means an equilateral triangle, a square, a regular pentagon, a regular hexagon and so on. We need not consider heptagons as the number of sides is less than 7. For a regular geometric figure, the area is completely determined by the number of sides and the length of a side. So we just need to find out if an area of 16 units corresponds to a side of integer length for the 4 regular geometric figures.

As it turns out, only the square can have both, an integral side and an integral length.

This is highly unlikely to be asked in the GRE, mainly because you need to separately calculate an expression for the area of a regular pentagon using some trigonometry. The takeaway is that counting problems can be deceptively simple but there are a lot of factors involved.

Permutation problems seem easier, mainly because it is clear that there is some sequence involved, some sort of ordering. But let’s try a hard permutation example:

Q: Consider the numbers from 1 to 10. How many ways is it possible to build a sequence of 5 numbers such that all the numbers are in ascending order?

This is actually a combination problem masquerading as a permutation problem. Given a set of distinct numbers, there is only one way to arrange all the numbers in ascending order. So what the question is really asking is the number of ways to select 5 numbers from a set of 10 numbers.

(Another variation: ‘How many orderings of 5 numbers are possible such that the second number is the largest number while the other numbers are in ascending order?’ Here, too, for a given set of numbers, this type of ordering is uniquely determined. So it’s just another combination problem.)

At a higher level, permutation and combination problems are deceptively simple. They are often conceptually harder than algebra and especially, geometry problems. Also, they are inherently open-ended, as they can involve concepts from algebra, geometry or other areas. So it’s really not enough to just know the factorial formulas, it helps if you have an intuition of what exactly is required.


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