Solve the equation \((n-4)\,{}^{n+1}C_5={} ^{n+2}C_7\).

The number of permutations of \(n\) items taken 4 at a time is \(\frac16\) times the number of permutations of \(2n\) items taken 3 at a time.

(a) Show that \(3n^2-19n+20=0\).

(b) Hence find the value of \(n\).

Given that \(\frac{{ }^{n+1} \mathrm{P}_{5}}{437}={ }^{n} \mathrm{P}_{3}\), use an algebraic method to find the value of \(n\).

(a) A 5-digit number is to be formed using 5 different numbers selected from \(1,2,3,4,5,6,7,8\) and 9 . No digit may be used more than once in any 5 -digit number. (i) Find how many 5-digit numbers can be formed.

(ii) Find how many of these 5-digit numbers are greater than 50000 and even.

(b) A team of 9 people is to be chosen from 6 doctors, 4 dentists and 2 nurses. Find how many possible teams include at least 2 doctors, at least 2 dentists and at least 2 nurses.

(a) A 5-character password is to be formed from the following 13 characters.

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

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

(ii) Find the number of possible passwords that contain at least one symbol.

(b) Given that

\(16\binom n{12}=(n-10)\binom{n+1}{11},\)

find the value of \(n\).

(a) A 4-digit code is made using digits chosen from \(0,1,2,\ldots,9\). No digit may be repeated, but the code may begin with 0.

Find the number of possible codes if

(i) there are no other restrictions,

(ii) the code is odd,

(iii) the code is greater than 1000.

(b) A team of 9 people is chosen from a group of 15 people. Four members of one family are in the group. The family members must not be separated: either all four are chosen or none of them is chosen.

Find the number of possible teams.

A photographer takes 12 different photographs. There are 3 of sunsets, 4 of oceans, and 5 of mountains.

(a) The photographs are arranged in a line on a wall.

(i) How many possible arrangements are there if there are no restrictions?

(ii) How many possible arrangements are there if the first photograph is of a sunset and the last photograph is of an ocean?

(iii) How many possible arrangements are there if all the photographs of mountains are next to each other?

(b) Three of the photographs are to be selected for a competition.

(i) Find the number of different possible selections if no photograph of a sunset is chosen.

(ii) Find the number of different possible selections if one photograph of each type, sunset, ocean and mountain, is chosen.

(a) The digits \(1,2,3,5,7,8\) are used to make 5-digit numbers. Each digit can be used at most once in each number.

(i) How many different numbers can be made?

(ii) How many of these numbers are not divisible by 5?

(iii) How many of these numbers are even and greater than \(30000\)?

(b) Given that

\({}^nC_3=6\,{}^nC_2,\)

find the constant \(n\).

A 4-digit code is formed using 4 different numbers chosen from \(1,2,3,\ldots,9\).

(a) Find the number of different codes that can be formed with no restrictions.

(b) Find the number of different codes that can be formed using only prime numbers.

(c) Find the number of different codes that can be formed if the first two numbers are even and the last two numbers are odd.

(d) Find the number of different codes that can be formed if the code is an even number.

A band can play 25 different pieces of music. From these pieces of music, 8 are to be selected for a concert.

(i) Find the number of different ways this can be done.

The 8 pieces of music are then arranged in order.

(ii) Find the number of different arrangements possible.

The band has 15 members. Three members are chosen at random to be the treasurer, secretary and agent.

(iii) Find the number of ways in which this can be done.

(a) Four parts in a play are to be given to four of the girls chosen from the seven girls in a drama class. Find the number of different ways in which this can be done.

(b) Three singers are chosen at random from a group of \(5\) Chinese, \(4\) Indian and \(2\) British singers. Find the number of different ways in which this can be done if

(i) no Chinese singer is chosen,

(ii) one singer of each nationality is chosen,

(iii) the three singers chosen are all of the same nationality.