The hawaiian alphabet known as the piapa was first written b

The hawaiian alphabet (known as the piapa) was first written by 19th century missinaries 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 5- letter words?

b) what is the maximum possible number of 8-letter words in which no letters are repeated?

c) How many 7-letter words can start with a U, end with a P, and contain no k\'s?

d) how many distinct arrangements are there of the letters in KAIKAMAHINE?

Solution

(a) Each place of the 5 - Letter can be filled in 12 ways

=> 12 12 12 12 12 = 12⁵ ways = 248832 ways

(b) Now , each place of the 8-Letter word has to be distinct

=> 12 11 10 9 8 7 6 5 = ( 12! )/( 4! ) ways = 19958400 ways

(c) As Middle letters can be occupied by any alphabet except K , each will have 11 ways to be filled

and start and end has to be with U and P respectively so only one way possible .

=> U 11 11 11 11 11 P = 11⁵ ways = 161051 ways

(d) KAIKAMAHINE is a 11 letter word with

2 K , 3 A , 2 I , 1 M , 1 H , 1 N and 1 E

Permutations of more than one same letters would be neglected by dividing by arrangement of those letters

=> Total Arrangements possible => 11!/( 2! * 3! * 2! ) = 1663200 ways