**1, 2, 3, 4, 5**

How many ways can we choose 2 numbers from the above 5, **without replacement**, when **the order in which we choose the numbers is important?**

\(5^{(2)}\) = \(\frac{5!}{(5-2)!}\) = 20

A pin number of length 4 is formed by randomly selecting and arranging 4 digits from the set {0,1, 2, 3,. . . 9} with replacement.

Find the probability of the event:

**A:** The pin number is even.

**B:** The pin number has only even digits.

**C:** All of the digits are unique.

**D:** The pin number contains at least one 1.

**P(A)** = \(\frac{5 * 10^3}{10^4}\)

**P(B)** = \(\frac{5^4}{10^4}\)

**P(C)** = \(\frac{10*9*8*7}{10^4}\) = \(\frac{10^{(4)}}{10^4}\)

**P(D)** = \(\frac{10^4-9^4}{10^4}\)

Any **Unordered (order does not matter)** sequence of k objects taken from a set of n distinct objects is called a **combinations** of size k of the objects denoted

\(C^k_n = {n \choose k} = \frac{n!}{k*n-k!}\)

Read (n choose k)
For n and k both non-negative integers with n ≥ k.

**Example**
Suppose there are 8 students in a group and that 5 of them must be selected to form a basketball team.

(a) How many different teams could be formed?

Use the combination rule with n = 8 and k = 5 as shown below:

\({8 \choose 5}\)

\(\frac{8!}{5!3!}\)

= 56

**Example**

A committee of 3 is to be formed from a group of 20 people. How many different committees are possible?

\({20 \choose 3}\)

\(\frac{20!}{3!17!}\)

= 1140