[12 points] Let $N = \{ 1,2,3,4,5,6,7,8,9 \}$ and $L = \{ a , b , c \}$.
(i) Suppose we arrange the 12 elements of $L \cup N$ in a line such that no two of the three letters occur consecutively. If the order of the letters among themselves does not matter, find the number such arrangements.
(ii) Find the number of functions from $N$ to $L$ such that exactly 3 numbers are mapped to each of $a , b$ and $c$.
(iii) Find the number of onto functions from $N$ to $L$.

Among 12-character strings made using all eight $a$'s and four $b$'s, how many strings satisfy all of the following conditions? [4 points]
(a) $b$ cannot appear consecutively.
(b) If the first character is $b$, then the last character is $a$.
(1) 70
(2) 105
(3) 140
(4) 175
(5) 210

When arranging $1, 2, 2, 4, 5, 5$ in a line to form a six-digit natural number, find the number of natural numbers greater than 300000. [4 points]

Among 12-character strings made using all eight $a$'s and four $b$'s, how many strings satisfy all of the following conditions? [4 points] (가) $b$ cannot appear consecutively. (나) If the first character is $b$, then the last character is $a$.
(1) 70
(2) 105
(3) 140
(4) 175
(5) 210

When arranging 5 white flags and 5 blue flags in a line, how many ways are there to place white flags at both ends? (Note: flags of the same color are indistinguishable from each other.) [3 points]
(1) 56
(2) 63
(3) 70
(4) 77
(5) 84

Let 10 red balls and 10 white balls be arranged in a straight line such that 10 each are on either side of a central mark. The number of such symmetrical arrangements about the central mark is
(A) $\frac { 10 ! } { 5 ! 5 ! }$
(B) $10 !$
(C) $\frac { 10 ! } { 5 ! }$
(D) $2 \cdot 10 !$

16. How many different nine digit numbers can be formed from the number 223355888 by rearranging its digits so that the odd digits occupy even position :
(A) 16
(B) 36

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(C) 60
(D) 180

The number of six letter words (with or without meaning), formed using all the letters of the word 'VOWELS', so that all the consonants never come together, is

If the arrangement of letters in a word from left to right is the same as from right to left, this word is called a palindrome word.
For example; NEDEN is a palindrome word.
Engin will create a 5-letter palindrome word using each of 3 distinct vowels and 4 distinct consonants as many times as he wants. In this word, two vowels should not be adjacent and two consonants should not be adjacent either.
Accordingly, how many different palindrome words can Engin create that satisfy these conditions?
A) 72 B) 84 C) 96 D) 108 E) 120