Given \({}^n{C_5} = {}^n{C_3}\), find \(n\).

On an examination paper, there are 6 calculus questions and 4 geometry questions. In how many ways can 6 questions be chosen if exactly 4 calculus questions are chosen?

A team of 4 is to be chosen from 5 boys and 6 girls. How many teams are possible if 2 girls are excluded? 

A team of 4 is to be chosen from 5 boys and 6 girls. How many teams are possible if there is at least 1 girl in the team? 

A class consists of 8 girls and 6 boys. How many ways are there of selecting a committee of 3 girls and 2 boys?

From the seven colours of the spectrum, four are to be selected.

i) How many different selections can be made?

ii) How many different selections can be made if the selection must not include red and blue?

From a group of nine people five are selected.

How many different groups of five may be selected so as to:

i) not include the youngest and the oldest 

ii) include the youngest and the oldest

iii) include the youngest or oldest but not both?

Prove that \({}^{n + 1}{C_{r + 1}} = {}^n{C_r} + {}^n{C_{r + 1}}\)

i) In how many ways can 4 persons be grouped into two pairs to play a set of doubles tennis?

ii) Eight tennis players are to play two simultaneous sets of doubles tennis on two separate tennis courts. In how many different ways can the tennis players be selected for these two sets of tennis?

i) In how many ways can 12 presents be divided into 4 piles of 3 presents?

ii) In how many ways can 12 presents be divided between 4 children, if each child is to receive the same number of presents?

Eleven people gather to play five a side soccer by forming two teams of five to play each other, with the remaining person acting as referee.

i) In how many different ways can the teams be formed?

ii) If two particular people are not to be in the same team, how many different ways are there to chose the teams?

A class of \(3n\) students is to be divided into three groups consisting of \(n-1\), \(n\) and \(n+1\) students.

i) Show that the number of ways that this can be done is \[\frac{(3n)!}{n!(n-1)!(n+1)!}\]

ii) Suppose that the three groups have been chosen. In how many ways can the \(3n\) students be arranged around a circular table if the students in each group are to be seated together? Give your answer in the factorial form.

A bag contains 3 red and 4 white marbles. Three of them are drawn at random and placed in a second bag. If two of them are then drawn at random from the second bag, what is the probability that they are both white?

Eight people arrive at a restaurant for dinner. They are provided with two circular tables, one of which seats 5 people and the other, 3.

i) How many different seating arrangements are there?
ii) If the seatings were arranged at random, what is the probability that a particular couple find themselves at separate tables?

A standard pack of 52 cards consists of 13 cards of each of the following suits: spades, hearts, clubs and diamonds.

i) In how many ways can five cards be selected without replacement so that exactly two are hearts and three are spades? (Assume that the order of selection of the five cards is not important).
ii) In how many ways can five cards be selected without replacement if at least four cards must be of the same suit? (Assume that the order of selection of the five cards is not important).

(i) In how many ways can a group of four people be divided into:

(\(\alpha\)) a group of three and a single person?
(\(\beta\)) two groups of two?

(ii) In how many different ways can a group of \(2n\) people be divided into two groups of \(n\) people?

There are eight points marked on the circumference of a circle. Using these points,

(i) how many different chords can be drawn ?
(ii) how many different triangles can be drawn?

In how many ways can a group of six people be divided into two unequal groups?

There are 5 red, 3 blue and 2 green balls in a bag. In how many ways can 3 balls be withdrawn

(i) with no restrictions?
(ii) if there are 2 red balls and 1 green ball?
(iii) if there is one of each colour?

On an examination paper, there are 4 questions on algebra and 6 questions on geometry. In how many ways can 5 questions be chosen if

(i) exactly 3 are algebra questions ?
(ii) at most 3 are algebra questions?
(iii) at least 3 are algebra questions?

A committee is to be appointed in a club containing 12 members. How many committees are possible, if each committee is to have

(i) 5 members
(ii) 7 members ?

Prove that \({ }^n C_r=\dfrac{n-r+1}{r} \times{ }^n C_{r-1}\)

The diagram shows a billard table with 9 balls lying on the table, 4 of which all lie in a straight line. No other group of 3 balls are collinear.

How many different triangular patterns can be formed using the 9 points as vertices?

Committees of 5 persons are to be selected from a group of 6 women an 4 men. How many committees are possible if

(i) they include 3 women and 2 men
(ii) there is a majority of women?

There are fourteen netball players in a team. Find how many ways:

(i) The starting seven can be chosen
(ii) they can be split into two teams of seven.

Consider the word COMBINE;

(i) what is the total number of combinations of the letters taken three at a time?
(ii) how many three letter combinations contain only two vowels?
(iii) how many three letter combinations contain the letter ' \(\mathrm{B}\) ' ?

Prove that \({ }^{n+2} C_k={ }^n C_k+2{ }^n C_{k-1}+{ }^n C_{k-2}\)

How many diagonals can be drawn in a hexagon?

Eighteen people arrive at a restaurant. There is one table for eight, one table for six and one table for four. In how many ways can they be assigned to a table.

In how many ways can a committee of 4 liberals and 5 labor MPs be chosen from 10 MPs of each party?

An icosagon is a polygon with 20 vertices Using these vertices,

(i) how many different chords can be drawn ?
(ii) how many different triangles can be drawn ?
(iii) how many different diagonals can be drawn ?

From three apples, three pears and two bananas, how many selections of four pieces of fruit can be made taking at least one of each kind?

A team of four is to be chosen from seven women. What is the probability that

(i) two particular women are in the team?
(ii) two particular women are not in the team?

How many ways can 12 different objects be divided

(i) into groups of six, four and two ?
(ii) into equal groups of four ?

(i) Prove that \({ }^n C_k={ }^n C_{n-k}\)

(ii) Hence find \(n\) if \({ }^n C_5={ }^n C_3\)

The ratio of the number of combinations of \((2 n+1)\) different objects taken \(n\) at a time to the number of combinations of \((2 n-1)\) different objects taken \(n\) at a time is \(26: 7\). Find the value of \(n\).

Prove that \({ }^{m+n} C_k={ }^m C_2+{ }^m C_1 \times{ }^n C_1+{ }^n C_2\)