Six different digits are chosen from the nine digits \(1,2,3,4,5,6,7,8,9\).
These digits are used to form a 6-digit number.
Find how many 6-digit numbers can be formed in the following cases.
(a) There are no restrictions.

(b) The number is greater than 700000 .

(c) The number is greater than 750000 .

A 6-character password is to be formed from the letters \(b,f,g,k,m\), the numbers \(3,5,7,9\), and the symbols *, !, @. Each character can be used at most once.

(a) Find the number of 6-character passwords with no further restrictions.

(b) Find the number of 6-character passwords that start and end with a symbol.

(c) Find the number of 6-character passwords that start with either a symbol followed by a number, or a number followed by a symbol, and end with 2 letters.

(a) A 4-digit number is to be formed using the digits \(0,2,4,5,6\) and 8 . The 4-digit number must not start with 0 . Any digit may be used at most once in the 4-digit number. (i) Find how many 4-digit numbers can be formed.

(ii) Find how many even 4-digit numbers can be formed.

(iii) Find how many 4-digit numbers that are divisible by 5 can be formed.

(b) Solve the equation \((n+1) \times{ }^{n+1} \mathrm{C}_{12}=33(n-10) \times{ }^{n} \mathrm{C}_{10}\).

(a) Five of the digits 123456789 are used to make a 5-digit number. Find how many ways this can be done when the 5 -digit number is greater than 50000 .

(b) In a group of 13 people, 7 have red hair and 6 have brown hair. (i) The 13 people stand in line for a photograph.

No person with red hair is standing next to another person with red hair. Find the number of different ways this can be done.

(ii) Chris chooses 5 people from this group.

Find the number of ways Chris can do this if at least 1 person chosen has brown hair.

(a) There are 3 girls and 2 boys standing in a straight line. Find the number of possible orders in each of the following cases. (i) No girls are next to each other.

(ii) The 2 boys are not next to each other.

(b) 12 people, including Anjie and Bubay, are divided into 3 groups of 4 people. Anjie and Bubay must not be in the same group.

Find the number of ways in which the 3 groups can be selected.

There are 3 women, 2 men and 4 children in a choir. (a) The choir stands in a single straight line. (i) Find the number of possible arrangements if the first person and last person are both women.

(ii) Find the number of possible arrangements if all the children stand next to each other.

(b) Four of the choir are selected to sing in a group. (i) Find the number of different selections if no man is chosen.

(ii) Find the number of different selections if at least 2 women are chosen.

Find the number of different ways the 9 letters of the word POLYMATHS can be arranged when (a) the O and A are not next to each other (b) the letters MATHS are together in this order.

A group of students, \(4\) girls and \(3\) boys, stand in line.

(a) Find the number of different ways the students can stand in line if there are no restrictions.

(b) Find the number of different ways the students can stand in line if the \(3\) boys are next to each other.

(c) Cam and Dea are \(2\) of the girls. Find the number of ways the students can stand in line if Cam and Dea are not next to each other.

A \(6\)-character password is to be formed from the following characters.

Letters: \(A,\ B,\ C,\ D\)

Numbers: \(1,\ 2,\ 3,\ 4\)

Symbols: \(*,\ \#,\ \$,\ \pounds\)

No character may be used more than once in any password.

(a)(i) Find the number of different \(6\)-character passwords that can be formed.

(a)(ii) How many of these \(6\)-character passwords end with a symbol?

(b) Find the number of different \(6\)-character passwords that include all the symbols, but do not start or end with a symbol.

A 5-digit code is to be formed using 5 different numbers selected from \(1,2,3,4,5,6,7,8\). Find how many possible codes there are if the code forms

(a) a number less than \(60000\) that ends in a multiple of \(3\),

(b) an even number less than \(60000\).

A 4-digit code is to be formed using 4 different numbers selected from \(2,3,4,5,6,7,8\) and \(9\). Find how many possible codes there are if the code forms

(a) a number that is odd and greater than \(5000\),

(b) a number greater than \(5000\) with a last digit that is prime.

(a) Eight books are to be arranged on a shelf. There are 4 mathematics books, 3 geography books and 1 French book.

(i) Find the number of different arrangements if there are no restrictions.

(ii) Find the number of different arrangements if the mathematics books have to be kept together.

(iii) Find the number of different arrangements if the mathematics books have to be kept together and the geography books have to be kept together.

(b) A team of 6 players is to be chosen from 8 men and 4 women. Find the number of different ways this can be done if

(i) there are no restrictions,

(ii) there is at least one woman in the team.

(a) Jack has won 7 trophies: 2 for cricket, 4 for football and 1 for swimming. Find the number of different arrangements if

(i) there are no restrictions,

(ii) the football trophies are to be kept together,

(iii) the football trophies are to be kept together and the cricket trophies are to be kept together.

(b) A team of 8 players is to be chosen from 6 girls and 8 boys. Find the number of different ways if

(i) there are no restrictions,

(ii) all the girls are in the team,

(iii) at least 1 girl is in the team.

(a) Eleven different television sets are to be displayed in a line. Of these, 6 are made by company \(A\) and 5 are made by company \(B\).

(i) Find the number of different ways the televisions can be arranged.

(ii) Find the number of different ways the televisions can be arranged so that no two sets made by company \(A\) are next to each other.

(b) A group of people is to be selected from 5 women and 3 men.

(i) Calculate the number of different groups of 4 people that have exactly 3 women.

(ii) Calculate the number of different groups of at most 4 people where the number of women is the same as the number of men.

A 5-digit code is formed using these characters.

No character can be repeated in a code. Find the number of possible codes if

(i) there are no restrictions,

(ii) the code starts with a symbol followed by two letters and then two numbers,

(iii) the first two characters are numbers, and no other numbers appear in the code.

A group of five people consists of two women, Alice and Betty, and three men, Carl, David and Ed.

(i) Three of these five people are chosen at random to be a chairperson, a treasurer and a secretary. Find the number of ways in which this can be done if the chairperson and treasurer are both men.

These five people sit in a row of five chairs. Find the number of different possible seating arrangements if

(ii) David must sit in the middle,

(iii) Alice and Carl must sit together.

A 7-character password is to be selected from the 12 characters shown in the table. Each character may be used only once.

Find the number of different passwords

(i) if there are no restrictions,

(ii) that start with a digit,

(iii) that contain 4 upper-case letters and 3 lower-case letters such that all the upper-case letters are together and all the lower-case letters are together.

A 7-character password is to be selected from the 12 characters shown in the table. Each character may be used only once.

Find the number of different passwords

(i) if there are no restrictions,

(ii) that start with a digit,

(iii) that contain 4 upper-case letters and 3 lower-case letters such that all the upper-case letters are together and all the lower-case letters are together.

A 5-digit code is to be formed from the digits \(1,2,3,4,5,6,7,8,9\). Each digit can be used once only in any code. Find how many codes can be formed if

(i) the first digit of the code is \(6\) and the other four digits are odd,

(ii) each of the first three digits is even,

(iii) the first and last digits are prime.

(a) A 5-character code is to be formed from the letters \(A,B,C,D,E,F\) and the numbers \(1,2,3,4,5,6,7\). Each character may be used once only in any code.

Find the number of different codes in which no two letters follow each other and no two numbers follow each other.

(b) A netball team of \(7\) players is to be chosen from \(10\) girls. \(3\) of these \(10\) girls are sisters. Find the number of different ways the team can be chosen if the team does not contain all \(3\) sisters.

(a) A 5-digit number is to be formed from the seven digits \(1,2,3,5,6,8,9\). Each digit can only be used once. Find the number of different 5-digit numbers if:

(i) there are no restrictions, (ii) the number is divisible by \(5\), (iii) the number is greater than \(60000\), (iv) the number is greater than \(60000\) and even.

(b) Ranjit has \(25\) friends, of whom \(15\) are boys and \(10\) are girls. He can only invite \(7\) friends. Find the number of ways the friends can be selected if:

(i) there are no restrictions, (ii) only \(2\) of the \(7\) friends are boys, (iii) the \(25\) friends include a boy and his sister who cannot be separated.