How many $3$ letter "words" consisting of at least $1$ vowel and $1$ consonant can be made from the letters of EQUATION? > The word EQUATION contains all five vowels. How many $3$ letter "words" consisting of at least $1$ vowel and $1$ consonant can be made from the letters of EQUATION? Hi, would anyone be able to check the answer for this? The answers say 540, but I keep getting 270. My means of working out were taking two cases: one with two vowels and one consonant, and one with one vowel and two consonants. Thanks

Here is another way - take all three letter words and deduct those containing just vowels or just consonants. This comes to $$8\cdot 7 \cdot 6-5\cdot 4 \cdot 3 - 3\cdot 2 \cdot 1=336-60-6=270$$

If letters are allowed to be repeated the number is $$8^3-5^3-3^3=512-125-27=360$$

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Another way of counting is to count the number of possibilities for each pattern of vowels and consonants.

$VVC: 5\times 4 \times 3=60$

$VCV: 5\times 3 \times 4=60$

$VCC: 5\times 3 \times 2=30$

$CCV: 3\times 2\times 5=30$

$CVC: 3\times 5\times 2=30$

$CVV: 3\times 2\times 5=60$

This gives $270$

I've done this longhand for clarity