Vigornii Alphabet

In the Vigornii alphabet there are 8 consonants and 4 vowels. How many different arrangements of 5 letters can be made if exactly 2 vowels must be used and no repetition of letters is allowed?
Source: NCTM Mathematics Teacher, February 2006

SOLUTION
1. Choose 2 places out of 5 places to put the 2 vowels
\binom{5}{2}=10
2. Put 2 vowels out of 4 vowels in the 2 chosen places
10\times 4\times 3=120
3. In the remaining 3 places put 3 consonants out of 8 consonants
120\times 8\times 7\times 6=40320
There are 40320 different arrangements.

Answer: 40320
Alternative solution
1. Choose 2 vowels out of 4 vowels
\binom{4}{2}=6
2. Choose 3 consonants out of 8 consonants
\binom{8}{3}=56
3. How many ways can we form a word with 2 vowels and 3 consonants?
6\times 56=336
4. For each of the 336 five-letter words how many different ways can we arrange the 5 letters?
336\times 5!=40320

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About mvtrinh

Retired high school math teacher.
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