(a) 3 men and 3 women are standing in a line. The 3 men are standing next to each other. Find how many different arrangements are possible.

(b) In the 13-letter word MULTIBRANCHED, there are 4 vowels, U, I, A and E. 7 different letters are selected from these 13 letters. Find how many different selections are possible if the selection includes at least 2 vowels.

In this question \(n\geqslant6\).

Use an algebraic method to show that \({}^{n}\mathrm{C}_5-{}^{n-1}\mathrm{C}_5\) can be written as \({}^{n-1}\mathrm{C}_4\).

A sports club has the following members. 5 runners, 4 swimmers, 3 gymnasts (a) These members stand in a straight line.

Find the number of ways that this can be done when all the runners stand together, all the swimmers stand together and all the gymnasts stand together.

(b) Four of these members are selected for an event.

Find the number of ways that this can be done when at least one runner, at least one swimmer and at least one gymnast must be selected.

(a) A team of 10 players is to be chosen from 15 players.

(i) Find the number of different teams that can be chosen if there are no restrictions.

The 15 players include 3 sisters who must not be separated.

(ii) Find the number of different teams that can be chosen.

(b) A 6-digit number is to be formed using the digits \(0,1,2,3,4,5,6,7,8\) and \(9\). The 6-digit number cannot start with \(0\) and all six digits must be different.

Find how many 6-digit numbers can be formed if the 6-digit number is even.

(a) A class contains 7 girls and 8 boys. A group of 6 is selected from the class. The group must contain at least 3 girls and at least 2 boys. Find the number of different groups that can be selected.

(b) A 5-character code is to be formed from the following characters.

No character may be used more than once in any code. The characters may be arranged in any order.

Find the number of different codes that can be formed using 4 letters and 1 number.

A class of 10 students includes Abby and Ben. (a) A group of 5 students is to be selected from the class. Find the number of possible groups in the following cases. (i) There are no restrictions.

(ii) The group includes both Abby and Ben.

(iii) The group includes either Abby or Ben, but not both.

(b) All 10 students are arranged in a line. How many arrangements are possible if there are exactly three students between Abby and Ben?

A team of 8 people is to be formed from 6 teachers, 5 doctors and 4 police officers. (a) Find the number of teams that can be formed. (b) Find the number of teams that can be formed without any teachers. (c) Find the number of teams that can be formed with the same number of doctors as teachers.

(a) A 6-digit number is to be formed using the digits \(0,1,2,3,4,5,6,7,8,9\). The 6-digit number cannot start with 0 . Each digit can be used at most once in any 6-digit number. Find how many of these 6 -digit numbers are divisible by 5 . (b) The number of combinations of \((n+1)\) objects taken 13 at a time is equal to 16 times the number of combinations of \(n\) objects taken 12 at a time. Find the value of \(n\).

(a) A team of 8 people is to be chosen from a group of 15 people.

(i) Find the number of possible teams.

(ii) Four members of the group are from the same family. The team must include either all four family members or none of them. Find the number of possible teams.

(b) Given that

\(\displaystyle (n+9)\times{}^nP_{10}=(n^2+243)\times{}^{n-1}P_9,\)

find the value of \(n\).

(a) Find the number of ways in which 14 people can be put into 4 groups containing 2, 3, 4 and 5 people.

(b) 6-digit numbers are to be formed using the digits \(0,1,2,3,4,5,6,7,8,9\). Each digit may be used only once in any 6-digit number. A 6-digit number must not start with 0. Find how many 6-digit numbers can be formed if

(i) there are no further restrictions,

(ii) the 6-digit number is divisible by 10,

(iii) the 6-digit number is greater than 500 000 and even.

(a)(i) A gardening group has 20 members. A committee of 6 members is to be selected. Two of the members, Ann and Bo, belong to the group, but at most one of them can be on the committee. Find the number of different committees that can be selected.

(a)(ii) A gate passcode has 6 characters. The characters are chosen from the letters and digits shown below, and no character is repeated.

Find the number of passcodes that consist of 4 letters followed by 2 digits.

(b)(i) Given that \(n\geq4\), show that

\((n-3)\binom n3=4\binom n4.\)

(b)(ii) Given that

\(\binom n3=5n,\qquad n\gt 3,\)

show that \(n^2-3n-28=0\), and hence find \(n\).

(a) A 6-digit number is to be formed using the digits \(0,1,2,3,4,5,6,7,8,9\). Each digit can be used only once in any 6-digit number. A 6-digit number cannot start with 0.

(i) Find how many 6-digit numbers can be formed.

(ii) Find how many of these 6-digit numbers are divisible by 5.

(b) A committee of 7 people is to be chosen from 6 doctors, 10 nurses and 8 dentists.

(i) Find the number of committees that can be chosen.

(ii) Find the number of committees that can be chosen if all the doctors have to be on the committee.

(iii) Find the number of committees that can be chosen if there has to be at least one dentist on the committee.

(a) A 5-character password is to be formed from the following 10 characters.

Letters: \(A,\ B,\ C,\ X,\ Y,\ Z\)

Symbols: *, $, #, &

No character can be used more than once in any 5-character password.

(i) Find the number of passwords that can be formed.

(ii) Find the number of passwords that can be formed if the password has to contain at least one symbol.

(iii) Find the number of passwords that can be formed if the password has to start with two letters and end with two symbols.

(b) A team of 8 people is to be chosen from 5 doctors, 4 teachers and 6 police officers. Find how many possible teams have the same number of doctors as teachers.

(a) A team of \(6\) people is to be chosen from \(10\) people. Two of the people are sisters who must not be separated. Find the number of different teams that can be formed.

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

Digits: \(2,4,8\). Letters: \(x,y,z\). Symbols: \(*,#,!\).

No character may be used more than once in any password. Find the number of different passwords that may be chosen if

(i) there are no other restrictions,

(ii) the password starts with two letters and ends with two digits.

(a) A \(6\)-digit number is formed from the digits \(0,1,2,5,6,7,8,9\). A number cannot start with \(0\) and each digit can be used at most once in any \(6\)-digit number.

(i) Find how many \(6\)-digit numbers can be formed if there are no further restrictions.

(ii) Find how many of these \(6\)-digit numbers are divisible by \(5\).

(iii) Find how many of these \(6\)-digit numbers are greater than \(850000\).

(b) A team of \(8\) people is to be chosen from \(12\) people. Three of the people are brothers who must not be separated. Find the number of different teams that can be chosen.

(a) A 5-digit number is to be formed using digits selected from 0, 1, 2, 3, 4, 5 and 6. No digit may be used more than once and the number may not start with 0.

(i) How many such 5-digit numbers can be formed?

(ii) How many of the numbers formed are even?

(b) A team of 7 people is to be selected from a group of 9 women and 6 men. Find the number of different teams that can be selected if the team must include at least one man.

(c) (i) Show that \({}^nC_3+{}^nC_2=\frac{1}{6}(n^3-n)\), where \(n\ge 3\).

(ii) Hence solve the equation \({}^nC_3+{}^nC_2=4n\), where \(n\ge 3\).

Given that

\(65\,{}^nC_5=2(n-1)\,{}^{n+1}C_6,\)

find the value of \(n\).

A group of \(15\) people includes \(3\) brothers. A team of \(6\) people is to be chosen from this group. The three brothers must not be separated. Find the number of possible teams that can be chosen.

Given that

\({}^n C_4=13\,{}^n C_2,\)

find the value of \({}^n C_8\).

(a)

(i) Find how many different 5-digit numbers can be formed using the digits \(1,3,5,6,8\) and \(9\). No digit may be used more than once in any 5-digit number.

(ii) How many of these 5-digit numbers are odd?

(iii) How many of these 5-digit numbers are odd and greater than \(60000\)?

(b) Given that

\(45\binom n4=(n+1)\binom{n+1}{5},\)

find the value of \(n\).

Five digits, \(1,3,5,8,9\), and three symbols, \(*,\$, \#\), are used to form passwords.

A password has six characters and no character is repeated.

(a) Find the number of different passwords which can be formed if the password

(i) has no restrictions,

(ii) starts with a digit and finishes with a digit,

(iii) starts with the three symbols.

(b) The number of combinations of \(5\) objects chosen from \(n\) objects is six times the number of combinations of \(4\) objects chosen from \(n-1\) objects.

Find \(n\).

(a) A committee of \(8\) people is to be formed from \(5\) teachers, \(4\) doctors and \(3\) police officers. Find the number of different committees that could be chosen if

(i) all \(4\) doctors are on the committee,

(ii) there are at least \(2\) teachers on the committee.

(b) Given that

\({}^nP_5=6\,{}^{\,n-1}P_4,\)

find the value of \(n\).

(a) Find how many different \(5\)-digit even numbers greater than \(50000\) can be formed using the digits \(0\), \(1\), \(4\), \(5\), \(6\), \(7\) and \(9\) if no digit is repeated in any number.

(b) Given that \(n\) is a positive integer and

\({}^nC_4=6{}^nC_2,\)

find the value of \(n\).

(a) A \(5\)-digit number is made using the digits \(0,1,2,3,4,5,6,7,8\) and \(9\). No digit may be used more than once in any \(5\)-digit number. Find how many such \(5\)-digit numbers are odd and greater than \(70000\).

(b) The number of combinations of \(n\) objects taken \(3\) at a time is \(2\) times the number of combinations of \(n\) objects taken \(2\) at a time. Find the value of \(n\).

Marc chooses \(5\) people from \(4\) men, \(4\) women and \(2\) children.

Find the number of possible selections when:

(a) there are no restrictions,

(b) at least \(2\) men are chosen,

(c) at least \(1\) man, at least \(1\) woman and at least \(1\) child are chosen.

(a)(i) Find how many different 4-digit numbers can be formed using the digits \(2\), \(3\), \(5\), \(7\), \(8\) and \(9\), if each digit may be used only once in any number.

(a)(ii) How many of the numbers found in part (i) are divisible by 5?

(a)(iii) How many of the numbers found in part (i) are odd and greater than \(7000\)?

(b) The number of combinations of \(n\) items taken 3 at a time is \(92n\). Find the value of the constant \(n\).

(a) In an examination a candidate must select 2 questions from the 5 questions in section A and 4 questions from the 8 questions in section B. Find the number of ways in which this selection can be made.

(b) The 7 digits of the number 6378129 are arranged to give a different 7-digit number. Find the number of different 7-digit numbers that can be made in which the number is even.

(a)(i) Find how many different 5-digit numbers can be formed using five of the eight digits \(1,2,3,4,5,6,7,8\), if each digit can be used once only.

(ii) Find how many of these 5-digit numbers are greater than \(60000\).

(b) A team of 3 people is to be selected from 4 men and 5 women. Find the number of different teams that could be selected which include at least 2 women.

(a)(i) Find how many different 4-digit numbers can be formed using the digits \(1,3,4,6,7\) and \(9\). Each digit may be used once only in any 4-digit number.

(a)(ii) How many of these 4-digit numbers are even and greater than \(6000\)?

(b) A committee of \(5\) people is to be formed from \(6\) doctors, \(4\) dentists and \(3\) nurses. Find the number of different committees that could be formed if

(i) there are no restrictions,

(ii) the committee contains at least one doctor,

(iii) the committee contains all the nurses.

(a) Twelve people are to be divided into three groups containing \(3\), \(4\) and \(5\) people. Find the number of possible divisions.

(b) Four-digit numbers are to be formed using four of the digits \(2,3,7,8,9\), with no digit repeated.

(i) Find the total number of such four-digit numbers.

(ii) Find the number of such four-digit numbers which are even.

(iii) Find the number of such four-digit numbers which are greater than \(7000\) and odd.

(a) Jess wants to arrange \(9\) different books on a shelf. There are \(4\) mathematics books, \(3\) physics books and \(2\) chemistry books. Find the number of different possible arrangements of the books if

(i) there are no restrictions,

(ii) a chemistry book is at each end of the shelf,

(iii) all the mathematics books are kept together and all the physics books are kept together.

(b) A quiz team of \(6\) children is to be chosen from a class of \(8\) boys and \(10\) girls. Find the number of ways of choosing the team if

(i) there are no restrictions,

(ii) there are more boys than girls in the team.

(a) A 5-digit code is to be chosen from the digits \(1,2,3,4,5,6,7,8\) and \(9\). Each digit may be used only once in any 5-digit code. Find the number of different 5-digit codes that may be chosen if

(i) there are no restrictions,

(ii) the code is divisible by \(5\),

(iii) the code is even and greater than \(70000\).

(b) A team of \(6\) people is to be chosen from \(8\) men and \(6\) women. Find the number of different teams that may be chosen if

(i) there are no restrictions,

(ii) there are no women in the team,

(iii) there are a husband and wife who must not be separated.

A squad of \(20\) boys, which includes \(2\) sets of twins, is available for selection for a cricket team of \(11\) players.

Calculate the number of different teams that can be selected if

(i) there are no restrictions,

(ii) both sets of twins are selected,

(iii) one set of twins is selected but neither twin from the other set is selected,

(iv) exactly one twin from each set of twins is selected.

(a) A football club has \(30\) players. In how many different ways can a captain and a vice-captain be selected at random from these players?

(b) A team of \(11\) teachers is to be chosen from \(2\) mathematics teachers, \(5\) computing teachers and \(9\) science teachers. Find the number of different teams that can be chosen if

(i) the team must have exactly \(1\) mathematics teacher,

(ii) the team must have exactly \(1\) mathematics teacher and at least \(4\) computing teachers.

(a) How many \(5\)-digit numbers are there that have \(5\) different digits and are divisible by \(5\)?

(b) A committee of \(8\) people is to be selected from \(9\) men and \(5\) women. Find the number of different committees that can be selected if the committee must have at least \(4\) women.

(a) Ten people are to be chosen, to receive concert tickets, from a group of 8 men and 6 women.

(i) Find the number of different ways the 10 people can be chosen if 6 of them are men and 4 of them are women.

The group of 8 men and 6 women contains a man and his wife.

(ii) Find the number of different ways the 10 people can be chosen if both the man and his wife are chosen or neither of them is chosen.

(b) Freddie has forgotten the 6-digit code that he uses to lock his briefcase. He knows that he did not repeat any digit and that he did not start his code with a zero.

(i) Find the number of different 6-digit numbers he could have chosen.

Freddie also remembers that his 6-digit code is divisible by 5.

(ii) Find the number of different 6-digit numbers he could have chosen.

Freddie decides to choose a new 6-digit code for his briefcase once he has opened it. He plans to have the 6-digit number divisible by 2 and greater than 600000, again with no repetitions of digits.

(iii) Find the number of different 6-digit numbers he can choose.

(a) A 6-digit number is formed using each of the digits \(1\), \(3\), \(5\), \(6\), \(8\), and \(9\) once and only once. Find the number of different 6-digit numbers that can be formed if

(i) there are no restrictions,

(ii) the number is even,

(iii) the number is even and greater than \(300000\).

(b) Ruby has \(15\) friends. She decides to invite \(8\) of them to a party. Find the number of ways in which she can do this if

(i) there are no restrictions,

(ii) two of the \(15\) friends are twins who must not be separated.

Naomi is going on holiday and intends to read \(4\) books during her time away. She selects these books from \(5\) mystery, \(3\) crime, and \(2\) romance books. Find the number of ways in which she can make her selection in each of the following cases.

(i) There are no restrictions.

(ii) She selects at least \(2\) mystery books.

(iii) She selects at least \(1\) book of each type.