From 7 different consonants and 4 different vowels, how many words of 3 consonants and 2 vowels can be formed? (A word is any arrangement of the 5 chosen letters.)
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → 25200
First choose the letters (combinations), then arrange the 5 chosen letters (permutations); the counts are multiplied.
Ways to choose 3 consonants out of 7: \(^7C_3=35\)
Ways to choose 2 vowels out of 4: \(^4C_2=6\)
Ways to arrange the 5 chosen letters: \(5!=120\)
Total \(=35\times6\times120=25200\)
Why the other options are incorrect
Option (1) 21020 — Incorrect. It is not a multiple of \(5!=120\), so it cannot count arrangements of 5 different letters.
Option (3) 10500 — Incorrect. It is not a multiple of \(5!=120\) (\(10500\div120=87.5\)), so it cannot count arrangements of 5 different letters.
Option (4) 21400 — Incorrect. It is not a multiple of 120 either, so it does not come from arranging the 5 letters.