## An example:

We have 10 diamond cards and 3 spade cards whereby each of the cards containing the same shape is assigned to a different number. Find the probability that when I draw 3 cards, there are 2 diamonds and 1 spade cards.

## Method 1: Using P&C

Probability required =

## Method 2: Using probability (similar to tree diagram)

Probability required =

## A question students may have:

One common question a student may have is: why does Method 2 contain the “” while Method 1 doesn’t? To investigate, let us apply the definition of combination:

For Method 1:

which gives us the same answer as Method 2.

In Method 2:We are performing the experiment as if we are drawing 1 card at a time.

The reason why we need to multiply by 3 is because to obtain 2 diamonds and 1 spade, the following **events are different** even though the end product is the same:

a) diamond, followed by diamond, followed by spade

b) diamond, followed by spade, followed by diamond

c) spade, followed by diamond, followed by spade.

However using P & C, **for combination, we do not need to consider the order** of which card is drawn (or should I say the 3 events are already factored in). You can view it as we are taking 3 cards at one go instead of 1 at a time.