Example

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

Combinations

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

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