Session 3-permutaiton combination probability 排列组合和概率

1. PERMUTATION

Suppose n objects are to be ordered from 1st to nth, each order is called a permutation. Apply the multiplication principle to count the number of permutations of n objects, i.e. n(n-1)(n-2)(n-3)....(3)(2)(1), or n!, called n factorial.

e.g. Suppose that 10 students are going on a bus trip, and each of the students will be assigned to one of the 10 available seats. What is the number of possible different seating arrangements of the students on the bus?

Notice: n objects should be distinguishable and they are always ordered in a line. If the objects are not ordered in a line but in other shapes, such as a circle or a square, how to calculate the number of permutations?

e.g. Five students are going to sit around a table, how many arrangement can there be? (If the relative position of two students is the same, then we view it as one arrangement.)

formula: (n-1)!

If there are some objects are exactly the same, the number of permutations should be calculated in another way.

e.g. How many different five-letter words can be formed when all letters in the word ENTER are used each time.

formula: n!/(number of repeated objects)!

Suppose that k objects will be selected from a set of n objects, where k<=n, and the k objects will be placed in order from 1st to kth. The number of permutation is n(n-1)(n-2)....(n-k+1).

e.g. How many different five-digit positive integers can be formed using the digits 1, 2, 3, 4, 5, 6, and 7 if none of the digits can occur more than once in the integer?

2. Combination

Given the five letters A, B, C, D, and E, determine the number of ways of selecting 3 of the 5 letters, but unlike before, you do not want to count different orders for the 3 letters. i.e.

Note that n choose k is always equal to n choose n-k

e.g. You should choose 3-person committee from a group of 9 students. How many ways are there to do this?

The difference between permutation and combination is whether order is considered. If different order makes different arrangement, it is permutation, and if different order makes no difference, it is combination. Most problems require us to combine these two.

e.g. Suppose you want to choose 5 members from a group of 8 to watering 5 different gardens. How many ways to do this?

3. Permutation& combination

Multiplication Principle:

If we need N steps to do a job and in the first step we have M1 different ways, second M2, third M3...., then we apply multiplication principle to count the total ways of doing this job, i.e. N=M1*M2*M3....*Mn

Addition Principle:

If we have N different ways to do a job and in the first way we have M1 different ways, second M2, third M3..., then we apply addition principle to count the total ways of doing this job, i.e. N=M1+M2+M3...+Mn

Notice: In multiplication principle steps are interdependent, while in addition principle each way can independently make the job done.

General method to solve permutation & combination problems:

1. Understand what needs to be done

2. Decide whether to take steps or divide into different ways, or both. When taking steps, how many steps are there; when dividing into different ways, how many ways in total.

4. Make sure of the requirements in each step or way, permutation or combination, and calculate the number of arrangements in each step or each way.

5. Count the total ways using multiplication principle or addition principle.

3.1 Give priorities to special elements and places

e.g. How many different 5-digit odd numbers can be formed when choosing from 0, 1, 2, 3, 4, 5 and each number can be used only once? (key: 288)

3.2 Bundling Strategy

e.g. There are 7 students to stand in a straight line. If student A and B should always stand side by side, so should student D and G, how many ways to line those 7 students? (key: 480)

Remember to make permutation between bundling elements