Sci-fi for the New Year
⎩ ⎪ ⎪ ⎨ ⎪ ⎪ ⎧ A E I U ⎭ ⎪ ⎪ ⎬ ⎪ ⎪ ⎫ { K P } ⎩ ⎪ ⎪ ⎨ ⎪ ⎪ ⎧ A E I U ⎭ ⎪ ⎪ ⎬ ⎪ ⎪ ⎫ { R S } ⎩ ⎪ ⎪ ⎨ ⎪ ⎪ ⎧ A E I U ⎭ ⎪ ⎪ ⎬ ⎪ ⎪ ⎫ { L N } ⎩ ⎪ ⎪ ⎨ ⎪ ⎪ ⎧ A E I U ⎭ ⎪ ⎪ ⎬ ⎪ ⎪ ⎫ .
A television network develops another sci-fi series, featuring a galaxy far, far away with aliens from a strange, strange planet.
Each alien has a name according to the pattern above.
Additionally, no name may contain the same vowel four times.
For instance, valid names include A K I S A N U and E P I S I L A ; but A K A R A L A is invalid.
How many different valid alien names can the series feature at most?
The answer is 2016.
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2 solutions
this is great
Without any restriction we have 4 4 × 2 3 = 2 0 4 8 options to make alien's names. But we have a restriction, and that is "none of the vowels can be repeated four times". Now the question is; how many names can we make with the three vowels in? The answer is "all the ways to put the consonants with each four vowels" which is 2 3 ∗ 4 = 2 5 where the first factor is the forms to arrange the consonants and the second is the forms to get four vowels together. So we need to sustract these words from our first number, then 4 4 × 2 3 − 2 5 = 2 0 4 8 − 3 2 = 2 0 1 6 And happy new year!
For the vowels, the number of possibilities is N v = 4 4 − 4 = 2 8 − 2 2 ( = 2 5 2 ) . The subtraction of 4 takes care of the forbidden vowel combinations: A x A x A x A , E x E x E x E , I x I x I x I , U x U x U x U .
For the consonants, the number of possibilities is N c = 2 3 ( = 8 ) .
In total, the number of possibilities is N = N v ⋅ N c = ( 2 8 − 2 2 ) ⋅ 2 3 = 2 1 1 − 2 5 = 2 0 4 8 − 3 2 = 2 0 1 6 .