Page 6 - Pra U STPM 2022 Penggal 3 - Maths (T)

Mathematics Semester 3 STPM Chapter 2 Probability
Whenever we calculate permutation, product like this comes up very frequently. We represent this product
by the notation n!, which is read “n factorial”. Thus,
.
.
n! = n(n – 1)(n – 2) … 3 2 1
Note: By definition 1! = 1 and 0! = 1.

First permutation rule

The number of permutations of n distinct elements is n!.
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Example 4
2
There is a photo taking session in a birthday party. If 6 people line up for taking a photo, how many
different ways can they be arranged from the left to the right?

Solution: Any of the 6 people can be placed in the first position from the left. Once the first
position is taken, there are 5 people left for the second position. After the first two
positions are taken, there are only 4 people to choose for the third position, and so on.
It is observed that there is one less person to choose from each time a position
is taken. Thus, the number of ways the 6 people could be arranged is
= 6 × 5 × 4 × 3 × 2 × 1
= 6!
= 720
Thus, they can line up in any of the 720 possible ways.

Sometimes, we may consider only a certain number of elements in a set to be arranged in order instead
of all of them. For example, if you were to arrange ten students to sit in the first row but you only have
three chairs, how many ways the chairs could be occupied? Although this problem is slightly different from
previous examples, the approach in getting the solution is similar. Imagine there are three slots and the slots
are to be filled one at a time. Any of 10 students may fill the first slot. After the first student is selected, any
of 9 students may fill the second slot, and any of 8 students in the last slot.

Possible ways: 10 9 8

The number of possible ways of placing 3 of the 10 students to sit in the first row is = 10 × 9 × 8
= 720

10
10
10
This product is commonly denoted by the symbol P . So, we have P = 10 × 9 × 8 = 720. P is read as

3
3
3
“the number of permutations of 10 objects taken 3 at a time”.
n
In general, P represents the number of ways r elements being selected from a set of n elements and placing
r
them in order. By following the similar procedure as the above example, we have
n P = n(n – 1)(n – 2) … [n – (r – 1)] = n(n – 1)(n – 2) … (n – r + 1)
r 1444442444443
r factors
.
This expression can be simplified by multiplying (n – r)(n – r – 1) … 2 1 , which is just equal to 1.
… .
(n – r)(n – r – 1) 2 1
74

02 STPM Math(T) T3.indd 74 28/10/2021 10:21 AM