Arrangement refers to selecting r elements from n elements, and considering the order of the elements. The total number of permutations is represented by the symbol P(n, r), and its calculation formula is: P(n, r)=n! /(n-r)！
Combination refers to selecting r elements from n elements, regardless of the order of the elements. The total number of combinations is represented by the symbol C(n, r), and its calculation formula is: C(n, r)=n! /(r！ (n-r)！ )
Operation steps of arrangement
Here is an example to introduce the operation steps of arrangement. Suppose there are five people, we should choose three of them to form a row and find different arrangements.
Step 1: Determine the number n of elements and the number r of selected elements. In this case, n=5 and r=3.
Step 2: Calculate the total number of permutations P(n, r). According to the formula P(n, r)=n! /(n-r)！ P(5，3)=5！ /(5-3)! =60。
Step 3: List all possible combinations. In this example, all combinations are 5*4*3=60.
Step 4: For each combination mode, arrange them in order. In this example, the combination of three people in each group can be arranged in order, so there are 60 different arrangements.
Combined operation steps
The following example introduces the operation steps of the combination. Suppose there are five people, we have to choose three of them and find different combinations.
Step 1: Determine the number n of elements and the number r of selected elements. In this case, n=5 and r=3.
Step 2: Calculate the total number of combinations C(n, r). According to the formula C(n, r)=n! /(r！ (n-r)！ ), you can get c (5,3) = 5! /(3! 2! )= 10。
Step 3: List all possible combinations. In this example, all combinations are 5*4*3/3*2* 1= 10.
Step 4: Regardless of the order of elements, each combination is only counted once. In this example, the combination of three people in each group is only counted once, so there are 10 different combinations.
Application of permutation and combination
Permutation and combination are widely used in mathematics, especially in probability theory and statistics. The following are some application examples of permutation and combination:
The combination of 1. permutation can be used to calculate the probability. For example, how many different straight cards are there in a deck of playing cards?
2. permutation and combination can be used to calculate combinatorial optimization problems. For example, in a factory, there are many machines that can complete the same task, so how to choose the best machine combination to complete the task?
3. permutation and combination can be used to calculate the queuing problem. For example, in a restaurant, there are multiple waiters who can serve the guests, and the arrival time and departure time of the guests are random. How to arrange the working hours of waiters and minimize the waiting time of guests?