1. How many different arrangements are there for the pens?
In a classroom there is a whiteboard and a line of four holes for keeping the marker pens for use on the board.
How many different arrangements are there for the pens?
Hole 1
Hole 2
Hole 3
Hole 4
4 choices
3 choices
2 choices
1 choices
4 x 3 x 2 x 1 = 24 different arrangements for the pens
4! = 24
No. arrangements = n! = n(n – 1)(n – 2)…1

2. 3 Coin example: 3 coins are chosen from a bag
C1, C2 and C3
How many different ways are there of choosing them? C1 C2 C3
C1 and C2 have the same value!!!

3. How many different ways can you arrange:
5 bricks in a line, each of a different colour.
120 ways

4. How many arrangements are there of:
5 bricks in a line, where 3 of them are red and 2 are blue
10 ways

5. Classwork
Ex 1A p10

6. Binomial Expansion Revision
Consider the expansion of
Why does the expansion have the symmetry in its coefficients?

7. If we take one term from each of the brackets and multiply them together, the possible arrangements are:
ppp = p3
ppq = p2q
pqq = pq2
pqp = p2q
qqq = q3
qqp = pq2
qpp = p2q
qpq = pq2

8. We can use factorial notation to find the coefficients of each term:
p3q0
p2q1
p1q2
p0q3
In general, a term for this expansion can be written as

9. The general term for the binomial expansion
for r = 0, 1, …, n

10. Example 1.
Find the binomial expansion of (p + q)5
Term
Coefficient p5
p4q
p3q2
p2q3
pq4 q5

11. Example 1.
Find the term in the expansion of (p + q)12 with p7.
The required term will be of the form Kp7q5
K =
So the term = 792p7q5
Class work
Exercise 1B p12
Questions: 1 – 10
Home work
Complete any 6 questions from Ex1B