Rule of product
Introduction example
How many different outfits can Per-Ulf Heiskanen choose to wear in the morning when he has two pairs of socks, three pants and two shirts to choose from?
Let's form a picture of the situation, which illustrates the idea of the rule of product:
In the picture, S = pairs of socks, H = pants and P = shirts.
Each “route” in the picture corresponds to one outfit. There are a total of 12 outfits. By the rule of product, we multiply the number of alternatives on each step with each other
Example 1. Per-Ulf Heiskanen is an avid player of standard betting pools. On behalf of Per-Ulf, calculate the following problems. There are a total of 13 lines in a standard betting pools and you can choose from three options in each
1 = Home team wins
X = Tie
2 = Away team wins
a) How many different standard betting pools entries are possible?
b) With what probability does Per-Ulf bet everything correctly?
c) With what probability does Per-Ulf bet everything wrong?
d) With what probability does Per-Ulf bet at least one line correctly?
a) It is possible to make these standard betting pool entries
Answer: There are 1594323 entries.
b) Entries with all the answers correct is only one
Answer: The probability of getting everything right is about 0.00000063.
c) There are two possibilities to choose wrong in each row, so according to the rule of product, these are the entries with all wrong answers
The probability that no bet is correct
Answer: The probability of getting everything wrong is 0,0051
d) Solve the task using a complementary event
P ("at least one correct") = 1-P ("all wrong") ≈ 0.99
Answer: The probability of betting at least one line correct is about 0,99
Number of orders and factorial
If there are n elements in the set, then it can be arranged in the queue n! (read by a factorial of n) in a different way.
Example 2
There are five children in a family, two boys and the rest are girls.
a) How many different ways can the five children stand in a line?
b) How many lines can be formed where the girls are next to each other?
a)
When choosing the first child in the line, there are 5 options to choose from.
When the second child is selected in the line, there are 4 options left to choose from.
When selecting the third child in the line, there are 3 options left to choose from
When selecting the fourth child in the line, there are 2 options left to choose from
When the last child in the line, the fifth is selected, there is only one option left.
According to the rule of product, the number of options for each step is multiplied by each other
Answer: The children can line up in a total of 120 different ways.
b)
Let’s start by thinking about how many different ways it is possible for girls to be in a row. T = girl and P = boy. Different options. TTTPP, PTTTP and PPTTT. There are a total of three options.
The girls can be in 3! different orders and the boys in 2! different orders. In total, by the rule of product, there are 3 · 3! · 2! = 36 different orders.
Answer: The children can line up so that the girls are next to each other in 36 different ways.
Example 3
How many different ways are there to choose a chairman, a deputy chairman and a secretary to the board of Ylä-Härmälä municipality, when the municipal board has 20 members to choose from.
There are 20 options to choose a chairman
There are 19 options to elect a deputy chairman
There are 18 options to choose a secretary
By rule of product, there are a total of 20 · 19 · 18 = 6840 options.
Answer: There are a total of 6840 options.
Number of subsets
When elements are selected from a set so that their order does not matter, we speak of subsets, i.e. different combinations.
Introduction example
How many ways to choose a group of three students from among seven students.
7 students can be set to 7! different orders.
In the assignment we are only interested in choosing a group of three students, so the rest (7-3)! orders can be divided out of the total. So there are a total of 7! divided by (7-3)! orders for three students!
A group of three can be selected for 3! different ways. When talking about groups, the order of the choices doesn’t matter, so it’s possible to choose different groups of three people
Answer: There are 35 different ways to choose a group of three students.
Let us generalise the introductory example and calculate how many different subsets of r elements can be formed from a set with a total of n elements.
In the future, the previous problem is simply calculated as follows
n = total number of elements in the group
r = how many groups of elements are to be selected from the group of n elements.
The count is read n over r, i.e. in this situation seven over three. The answer is how many groups of three elements can be selected from the seven elements.
Number of subsets with a calculator.
Depending on the type of calculator, the number of subsets can be calculated, with the following commands
Using a CAS calculator the command is usually: nCr (n, r).
'Function' calculators have an 'nCr' key and the command chain goes: n + nCr key + r.
Check the result of the previous introductory example using a calculator.
Example 4
How many ways can a group of 3 students be selected from a group of 30?
n = 30 (how many members are there in the group)
r = 3 (how many member groups to choose)
Answer: From a group of 30 students, a group of three can be selected in 4060 different ways.
Example 5
What is the probability that you would get three aces in a poker game with your first hand? In poker, you are dealt five cards in the first hand.
How many ways to choose three of the four aces:
How many other cards can be selected:
In what ways can a group of five cards be selected in total:
Probability is calculated as the quotient of favourable five-card groups and all possible five-card groups. By the rule of product, these are the favourable groups
So the probability is
Answer: The probability of getting three aces on your first hand is about 0,0017
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