Three-Letter Word Constraint Analysis

The goal is to count the number of three-letter English words that meet specific criteria:

• Must contain at least one consonant.
• Must not contain two consecutive consonants.

Alphabet Vowel Consonant Counts

We define the sets of vowels and consonants in the English alphabet:

• Number of vowels (V): 5 (A, E, I, O, U)
• Number of consonants (C): 21 (All other letters)
• Total letters: 26

Valid Structures No Consecutive Consonants

For a three-letter word, the condition of having no two consecutive consonants eliminates structures like CCC, CCV, VCC. The possible valid structures are:

• CVC: Consonant - Vowel - Consonant
• CVV: Consonant - Vowel - Vowel
• VCV: Vowel - Consonant - Vowel
• VVC: Vowel - Vowel - Consonant

The structure VVV (all vowels) is considered but ruled out by the "at least one consonant" condition.

Calculating Word Counts per Structure

We calculate the number of possible words for each valid structure:

• CVC Structure: The number of possibilities is the product of the counts for each position. Calculation: $N(C) \times N(V) \times N(C) = 21 \times 5 \times 21 = 2205$.
• CVV Structure: Calculation: $N(C) \times N(V) \times N(V) = 21 \times 5 \times 5 = 525$.
• VCV Structure: Calculation: $N(V) \times N(C) \times N(V) = 5 \times 21 \times 5 = 525$.
• VVC Structure: Calculation: $N(V) \times N(V) \times N(C) = 5 \times 5 \times 21 = 525$.

Total Word Count Summation

To find the total number of words satisfying all conditions, sum the counts from the valid structures (CVC, CVV, VCV, VVC):

Total Words = $2205 + 525 + 525 + 525$

Total Words = $2205 + 1575 = 3780$.