8) The Hawaiian alphabet (known as the piapa) was first written by 19 th century

Question

8) The Hawaiian alphabet (known as the piapa) was first written by 19th century missionaries and consists of 12 letters; the vowels A, E, I, O, and U, and the consonants H, K, L, M, N, P, and W. Assuming that all possible arrangements of these letters could be words:

a) What is the maximum possible number of 4-letter words?

b) What is the maximum possible number of 7-letter words in which no letters are repeated?

c) How many 8-letter words can start with a P, end with an A, and contain no U’s?

d) How many distinct arrangements are there of the letters in KOLAUKALAKI?

Explanation / Answer

Counting

8) The Hawaiian alphabet (known as the piapa) was first written by 19th century missionaries and consists of 12 letters; the vowels A, E, I, O, and U, and the consonants H, K, L, M, N, P, and W. Assuming that all possible arrangements of these letters could be words:

Number of possible ways = 12P4 = 12! / (12 – 4)! = 12! / 8! = 11880

Maximum number of 4-letter words = 11880

Number of possible ways = 12C7 = 12! / (12 – 7)! * 7! = 12! / 5! * 7! = 792

Maximum possible words = 792

Word is a 8-letter word.

This word should not contain U, this means there are 11 letters remaining for selection.

First letter can be select in only one way.

Last letter can be select in only one way.

Therefore for the middle 6 letters, we have 9 letters for selection.

Number of ways for middle 6 letters = 9C6 = 84

Total number of ways = 84

Total letters = 11

Letter K repeated 3 times

Letter L repeated 2 times

Letter A repeated 3 times

Number of arrangements = 11! / 3! * 2! * 3!

Number of arrangements = 554400

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