Superlesson
Project 6-1

Advanced Algebra

Chapter 6, Counting and Arranging Discrete Objects

6-1, Counting Objects and Permutations

Part A, Data and Scatter Plots

Mathematics can be used to help analyze language. Permutations and combinations are important tools for breaking codes, but they can also be helpful in solving simple word games.

Part A, Methods of Counting

1. The 8-letter words equation, dialogue, and sequoia are the three shortest common words containing the 5 common vowels. But the 8-letter scientific names of a certain beetle and fly contain 6 vowels (a, e, i, o, u, y), plus two different consonants.

a. The beetle was discovered in 1842, the fly in 1911. Here is the pattern of the vowels (V) and the consonants (C) in their names:

How many choices of vowels are there for V₁? for V₂? for each of the other vowels?

b. How many choices of consonants are there for C₁? for C₂?

c. How many 8-letter words follow the vowel-consonant pattern of the two names?

d. Find the names of the beetle and the fly.

Part B, Arrangements and Permutations

2. In a World Championship Scrabble match, your letter rack contains these letters, all consonants.At the top of the board there's a free, to build a word around. You want to use all of your letters on this turn. To do this you'll have to think of the word which, in its plural form, is the longest word in the English language containing only a single vowel.

a. How many permutations of the letters T G S N R H T E will you have to consider to find the only one you can use?

b. What is the word? If you can't see it, consult an anagram engine.

Part C, Making Connections

3. Along with letter frequencies and word frequencies, double-letter patterns are extremely useful in cracking codes.

a. Find the most common double-letter patterns.

b. A biologist had to postpone an exhibit when her Gymnoti got sick. She hung a sign which, encoded in a substitution cipher, read MMPS BTT ZTT. How many 3-letter patterns containing a double letter are there? How many 4-letter words containing a double letter are there?

c. What did the sign say? Find out what a Gymnotus is.