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

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!!!

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

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

Classwork Ex 1A p10

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

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

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

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

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

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