The total number of words (not necessarily meaningful) that can be made from the letters a,...
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The total number of words (not necessarily meaningful) that can be made from the letters a, d, e, f, g, h, j, k taken 4 together such that each word contains at least one vowel when repetition is not allowed?
Show answer & explanation
We use complementary counting: total 4-letter permutations minus those with no vowels. From 8 letters (a, d, e, f, g, h, j, k), we have 2 vowels (a, e) and 6 consonants (d, f, g, h, j, k). Total permutations of 4 from 8 = P(8,4) = 1680. Permutations with NO vowels (all 4 from 6 consonants) = P(6,4) = 360. Words with at least one vowel = 1680 − 360 = 1320.
Step-by-step Derivation:
Step 1: Identify vowels and consonants.
- Vowels: a, e (2 vowels)
- Consonants: d, f, g, h, j, k (6 consonants)
- Total letters: 8
Step 2: Calculate total permutations of 4 letters from 8 (without restriction).
P(8,4) = 8!/(8−4)! = 8!/4! = 8 × 7 × 6 × 5 = 1680
Step 3: Calculate permutations with NO vowels (all 4 consonants from 6).
P(6,4) = 6!/(6−4)! = 6!/2! = 6 × 5 × 4 × 3 = 360
Step 4: Apply complementary counting.
Words with at least one vowel = Total − No vowels
= 1680 − 360 = 1320 ways