Society & Culture & Entertainment Education

Challenging Counting Problems

Counting can seem like an easy task to perform. As we go deeper into the area of mathematics known as combinatorics, we realize that we come across some large numbers. Since the factorial shows up so often, and a number such as 10! is greater than three million, counting problems can get complicated very quickly if we attempt to list out all of the possibilities.

Sometimes when we consider all of the possibilities that our counting problems can take on, it's easier to think through the underlying principles of the problem.

This strategy can take much less time than trying brute force to list out a number of combinations or permutations. The question "How many ways can something be done?" is a different question entirely from "What are the ways that something can be done?" We will see this idea at work in the following set of challenging counting problems.

The following set of questions involves the word TRIANGLE. Note that there are a total of eight letters. Let it be understood that the vowels of the word TRIANGLE are AEI, and the consonants of the word TRIANGLE are LGNRT.

  1. How many ways can the letters of the word TRIANGLE be arranged?
  2. How many ways can the letters of the word TRIANGLE be arranged if the first three letters must be RAN (in that exact order)?
  3. How many ways can the letters of the word TRIANGLE be arranged if the first three letters must be RAN (in any order)?
  4. How many ways can the letters of the word TRIANGLE be arranged if the first three letters must be RAN (in any order) and the last letter must be a vowel?


  1. How many ways can the letters of the word TRIANGLE be arranged if the first three letters must be RAN (in any order) and the next three letters must be TRI (in any order)?
  2. How many different ways can the letters of the word TRIANGLE be arranged if the order and the placement of the vowels IAE cannot be changed?
  3. How many different ways can the letters of the word TRIANGLE be arranged if the order of the vowels IAE cannot be changed, though their placement may (IAETRNGL and TRIANGEL are acceptable but EIATRNGL and TRIENGLA are not)?
  4. How many different ways can the letters of the word TRIANGLE be arranged if the order of the vowels IAE can be changed, though their placement may not?
  5. How many different ways can six letters of the word TRIANGLE be arranged?
  6. How many different ways can six letters of the word TRIANGLE be arranged if there must be an equal number of vowels and consonants?
  7. How many different ways can six letters of the word TRIANGLE be arranged if there must be at least one consonant?
  8. How many different ways can six letters of the word TRIANGLE be arranged if the vowels must alternate with consonants?
  9. How many different sets of four letters can be formed from the word TRIANGLE?
  10. How many different sets of four letters can be formed from the word TRIANGLE that have two vowels and two consonants?
  11. How many different sets of four letters can be formed from the word TRIANGLE if we want at least one vowel?

Leave a reply