Eight apples, divided among 3 people, each with at least 1 apple, how many divisions are there?

1 for the first person, 116,125,134,143,152,161 for the first person2, 215,224,233,242,251 for the first person3, 314,323,332,341 for the first person, 413,422,431

The first person was 5 and had 512,521

The first person has 6 and has 611

So there are a total of 6+5+4+3+2+1=21 divisions.

There are 21 divisions.

1 for the first person, 116,125,134,143,152,161 for the first person2, 215,224,233,242,251 for the first person3, 314,323,332,341 for the first person, 413,422,431

The first person was 5 and had 512,521

The first person has 6 and has 611

So there are a total of 6+5+4+3+2+1=21 divisions.

This problem applies the drawer principle.

One per person first.

Then five apples are divided into three people, two without 3 kinds, one without 3 x 4, and all three with 6 kinds, a total of 21 divisions.

You can have 2 per person, so that at least 1 per person is divided, and there are 2 left.

One per person is also available.

5 apples are divided among 3 people, and each person is divided into one less. The three people in the question are not divided into A, B and C, so there are two divisions:

2) 3, 1, 1, if divided into A, B, C, there can be 6 divisions:

1) There are: 2, 2, 1; 2，1，2:

5 apples can be divided.

or, 1+2=3

1=1, there are two divisions, and three people have 2, 2, 1;

or 3, 2, 1 respectively.

Dividing 9 apples into 3 people, at least one for each person, can be seen as a problem of 8 empty inserts into three gears between 9 apples, so there is a common method.

c³₈=8×7×6/(3！)=56

In the same way, the 27 apples are divided as:

c³₂₆26×25×24/6=2600

Solution analysis: This kind of problem can be regarded as the interpolation method.

For three people with nine apples. 3 apples per person.

A division: 9 3=3.

This is a linear programming problem:

Let A, B, and C be three, A gets x, B gets Y, and C gets (27-x-y), and then makes :

x 1, y 1, 27-x-y 1, the integer coordinates enclosed by three straight lines can be counted.

There are 9 divisions. First of all, each person was given one and there were two left. After that, there is the following division:

1. Both are assigned to the same person, and there are 3 divisions:

2. The remaining two apples are divided one by one:

a. Divide the first apple first: 3 divisions;

b. Divide the last one again, except for those who have been divided in a: 2 divisions;

A total of 3 times 2 = 6 divisions, 1 and 2 together for a total of 9 divisions.

There are 6 divisions: each child is given apples, a person has a minimum of 1 apple, a maximum of 3 apples, so that there are 1,2), (3,1,1), (1,3,1), (1,1,3), (1,1,3).

From n different elements, any different elements of m (m n, m and n are natural numbers, the same below) are arranged in a certain order, which is called taking out an arrangement of m elements from n different elements; The number of all permutations of m (m n) elements from n different elements is called the number of permutations of m elements from n different elements, which is represented by the symbol a(n, m).

Calculation formula: a(n,m)=n!/(n-m）！；In addition, it is stipulated that 0! = 1<>

There are 9 divisions. First of all, each person was given one and there were two left. After that, there is the following division:

1. Both are assigned to the same person, and there are 3 divisions:

2. The remaining two apples are divided one by one:

a. Divide the first apple first: 3 divisions;

b. Divide the last one again, except for those who have been divided in a: 2 divisions;

A total of 3 times 2 = 6 divisions, 1 and 2 together for a total of 9 divisions.

At least one apple, each person has 2 apples, and the different divisions depend on the apples of these 2 apples are given to one person, then there are 3 divisions;

If an apple is divided into 2 people, one person does not get it, and the same is 3 divisions.

Therefore, there are 6 divisions.

There are 3 divisions of 221.

There are also 3 divisions of 311.

There are 6 types in total.

a b c

There are three types of a that start with 1, and you can also start with 2 or 3.

3*3=9 (species).

There are 6 divisions: each child is given apples, a person has a minimum of 1 apple, a maximum of 3 apples, so that there are 1,2), (3,1,1), (1,3,1), (1,1,3), (1,1,3).

From n different elements, any different elements of m (m n, m and n are natural numbers, the same below) are arranged in a certain order, which is called taking out an arrangement of m elements from n different elements; The number of all permutations of m (m n) elements from n different elements is called the number of permutations of m elements from n different elements, which is represented by the symbol a(n, m).

Calculation formula: a(n,m)=n!/(n-m）！；In addition, it is stipulated that 0! = 1

70 species.

1. Use the partition method, put 9 apples well, select 4 of the 8 empty positions and put them into the partition, divide the 9 apples into 5 parts, at least 1 of each part total: c(8,4)=70 species.

The answer should be 70.

2. Taxonomy.

1 person gives 1 first.

Total 70

with the partition method.

Put 9 apples in place, there are a total of 8 empty positions.

Take 4 of them and put them on the divider, and divide 9 apples into 5 parts, each with at least 1 part in total: c(8,4) = 70 kinds.

Apples are all the same. But 5 people are different. So the different divisions refer to the difference in the number of apples each person gets.

Everyone has at least one apple, so give everyone an apple first.

The question becomes that if you divide 4 apples among 5 people, you can divide them at will, how many different ways are there to divide them.

The following is the case where a person gets a number of apples, followed by the calculation of the different divisions that may occur in that case.

4,0,0,0,0 -5 kinds.

3,1,0,0,0 -5x4 = 20 types2,2,0,0,0 - (5x4) (2x1)=10 types2,1,1,0,0 - (5x4x3) (2x1)=30 types1,1,1,1,0 -5 types.

The final answer is 5 + 20 + 10 + 30 + 5 = 70 kinds.

Number Column** Fruit 1 9 (ordinate) Person 1 5 (abscissa) Count by yourself.

There is no difference between apples, but there is a difference between people.

11-(2-1) 3=8, dao 11 apples are distributed to 3 people, each to.

2 less weight, equivalent to 8 apples divided among 3 people, at least 1 each.

With the plug-in method, 8 apples are arranged in a row, 7 empty between two and two, any 2 empty spaces can be divided into 3 parts, c(7,2)=21, there are 21 divisions.