University Mathematics Handbook (2015)

XV. Combinatorics and Newton's Binomial

Chapter 2. Variations

2.1  Variations without Repetition

a. Any choice of distinct objects out on given distinct objects, when the order of choice is significant, is called a variation.

b. One variation of distinct objects is different from another variation of distinct objects in the order of terms of a least in one term.

c. The number of all different variations of distinct objects out of distinct objects is denoted as and equals to

 (*)

d. Multiplication principle: In a given set of distinct objects:

1. There are possible choices for the first-place term.

2. For any of the first-place choices, there are just possible second-place choices, and therefore, there are possible choices of two distinct objects.

3. For any of these possible choices, there are possible third-place choices, and therefore, the number of possible different threesomes is .

4. Following the same principle, we will find formula (*) of calculating the number of different variations of out of distinct objects.

Example: How many numbers of different digits can you form using the digits ?

Solution: We arrange these digits in 5 positions. The first on the left can be digits, excluded. The second on the left can also be digits, including zero, but one digit had already been used. Third on the left can be digits, fourth on the left, digits, and the fifth on the left, digits. Following the multiplication principle, the number of possible numbers is .

2.2  Variations with Repetition

a. A variation with repetition of out of distinct objects is a choice of out on given distinct objects, when the order of choice is significant, and any object can be chosen several times.

b. The number of variation with repetition of out of objects is denoted as .

c. Using the multiplication principle, we get

d.  Example: How many numbers of not necessarily different digits can you form using the digits ?

Solution: Using the multiplication principle, we get