Cambridge Additional Mathematics

256 Counting and the binomial expansion (Chapter 10)

Opening problem

Things to think about:

a If each committee member shakes hands with every other

committee member, how many handshakes take place?

Can a 10 -sided convex polygon be used to solve this

problem?

b If all 273 delegates shake hands with all other delegates,

how many handshakes take place now?

c If the organising committee lines up on stage to face the

delegates in the audience, in how many different orders can

they line up?

TheOpening Problemis an example of acountingproblem.

The following exercises will help us to solve counting problems without having to list and count the

possibilities one by one. To do this we will examine:

² the product principle ² counting permutations ² counting combinations.

Suppose there are three towns A, B, and C. Four different

roads could be taken from A to B, and two different roads

from B to C.

How many different pathways are there from A to C going

through B?

If we take road 1 , there are two alternative roads to complete

our trip.

Similarly, if we take road 2 , there are two alternative roads

to complete our trip.

The same is true for roads 3 and 4.

So, there are 2+2+2+2=4£ 2 different pathways from A to C going through B.

Notice that the 4 corresponds to the number of roads from A to B and the 2 corresponds to the number of

roads from B to C.

THE PRODUCT PRINCIPLE

If there aremdifferent ways of performing an operation, and for each of these there arendifferent ways

of performing a secondindependentoperation, then there aremndifferent ways of performing the two

operations in succession.

The product principle can be extended to three or more successive independent operations.

A THE PRODUCT PRINCIPLE

A C

B

road 1

road 2

road 3

road 4

A C

B

At a mathematics teachers’ conference there are 273 delegates present. The organising committee consists

of 10 people.

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Additional Mathematics
Y:\HAESE\CAM4037\CamAdd_10\256CamAdd_10.cdr Friday, 4 April 2014 1:42:37 PM BRIAN