Imagine you are asked to choose 3 friends to join a futsal team. Does the order of selection matter? Of course not! What matters is who is selected, not the order in which they are chosen.
This is the fundamental difference between combination and permutation. Combination is a way to select a certain number of objects from a larger collection of objects, where order is not considered.
In daily life, we often encounter combinations when:
Choosing food menus from a list of options
Determining team members for an activity
Selecting elective subjects at school
Determining color combinations for design
The difference from permutation is very clear: if in permutation ABC is different from , then in combination is the same as because the members are the same, only the order is different.
To determine the number of ways to select k objects from n available objects, we use the combination formula:
C(n,k)=(kn)=(n−k)!×k!n!
Where:
C(n,k) or (kn) means the combination of k objects from n objects
n is the total number of available objects
k is the number of objects to be selected
n! is the factorial of n
Why is this formula different from permutation?
The combination formula actually comes from the permutation formula divided by k!:
C(n,k)=k!P(n,k)=(n−k)!×k!n!
Division by k! is done because in combinations, we don't care about order. Each group of k objects has k! different arrangement possibilities, but they are all considered the same in combinations.
From 10 available students, how many ways can we select 5 students for a basketball team?
C(10,5)=5!×5!10!
C(10,5)=5×4×3×2×110×9×8×7×6=12030.240=252 ways
Food Menu Selection:
A restaurant offers 8 types of food and you can choose 3 types. How many possible combination choices are there?
C(8,3)=5!×3!8!
C(8,3)=3×2×18×7×6=6336=56 combinations
Situations with Special Conditions:
Sometimes there are certain restrictions in selection. For example, from 12 students (7 males and 5 females), we must select 4 students with the condition of at least 2 females.
Solution strategy:
Count all possibilities that meet the conditions
Separate based on conditions:2 females+2 males, 3 females+1 male, or 4 females
From 8 different books, how many ways can you choose 3 books to read during vacation?
A futsal team has 12 players. How many ways can they select 5 players to play on the field?
In a box there are 6 red balls and 4 blue balls. How many ways can you take 5 balls with the condition of at least 3 red balls?
A student must choose 4 subjects from 10 available subjects. If 6 subjects are mandatory and 4 subjects are elective, how many ways can they choose if there must be at least 2 mandatory subjects?
