The Chemistry Maths Book, Second Edition

608 Chapter 21Probability and statistics

objects can therefore be arranged inn(n 1 − 1 1)ways. The third object can occupyn 1 − 12

positions, and the number of ways of arranging 3 different objects isn(n 1 − 1 1)(n 1 − 1 2).

In general:

2.The number of permutations of ndifferent objects taken rat a time is

(21.22)

The total number of permutations of nobjects (taken nat a time) is therefore

n

P

n

1 = 1 n!

EXAMPLE 21.8The permutations of the 4 objects A, B, C, D taken 2 at a time are

AB and BA, AC and CA, AD and DA, BC and CB, BD and DB, CD and CD

and

4

P

2

1 = 1 4! 2 (4 1 − 1 2)! 1 = 112.

0 Exercises 16, 17

Combinations

In a permutation, the order of the selected objects is important. A combination, on

the other hand, is a selection of objects without regard to order.

3.The number of combinations of ndifferent objects taken rat a time is

(21.23)

EXAMPLE 21.9The number of combinations of 5 objects taken 3 at a time is

Thus, for A, B, C, D, E:

ABC, ABD, ABE, ACD, ACE, ADE, BCD, BCE, BDE, CDE

There are a possible 3!permutations for each combination, so that3! 1 × 1

n

C

3

1 = 1

n

P

3

. In

general, as can be seen from (21.22) and (21.23),

n

P

r

1 = 1 r! 1 × 1

n

C

r

.

0 Exercise 18

5

3

5

32

54321

321 21

C = 10

!

!!

=

⋅⋅⋅⋅

⋅⋅ ⋅

=

()()

n

r

C

n

rn r

n

r

=

!

!−!

=





















()

n

r

Pnn n nr

n

nr

=− − −+=

!

−!

()( )( )

()

12  1