How many ways can the number 12 be expressed as a sum of pos

How many ways can the number 12 be expressed as a sum of positive integers, each of which is greater than or equal to 2, if the order of the positive integers matters? For example, 6 can be written in 5 ways: 6, 2+4, 3+3, 4+2, and 2+2+2.

Solution

12 can be expressed as

2+2+2+2+2+2

2+2+2+2+4

2+2+2+4+2

2+2+4+2+2

2+4+2+2+2

4+2+2+2+2

2+2+2+3+3

2+2+3+2+3

2+3+2+2+3

3+2+2+2+3

3+2+2+3+2

3+2+3+2+2

3+3+2+2+2

2+2+3+3+2

2+3+3+2+2

2+3+2+3+2

2+2+2+6

2+2+6+2

2+6+2+2

6+2+2+2

2+2+4+4

2+4+2+4

4+2+2+4

4+2+4+2

4+4+2+2

2+4+4+2

3+3+3+3

2+4+3+3

4+2+3+3

2+3+4+3

4+3+2+3

3+2+4+3

3+4+2+3

3+2+3+4

3+4+3+2

3+3+2+4

3+3+4+2

2+3+3+4

4+3+3+2

2+2+3+5

2+2+5+3

3+5+2+2

5+3+2+2

2+3+2+5

2+5+2+3

2+3+5+2

2+5+3+2

3+2+2+5

5+2+2+3

3+2+5+2

5+2+3+2

2+2+8

2+8+2

8+2+2

2+3+7

2+7+3

7+2+3

7+3+2

3+2+7

3+7+2

2+4+6

2+6+4

4+2+6

4+6+2

6+2+4

6+4+2

2+5+5

5+2+5

2+5+5

3+3+6

3+6+3

6+3+3

3+4+5

3+5+4

4+3+5

4+5+3

5+3+4

5+4+3

4+4+4

2+10

10+2

3+9

9+3

4+8

8+4

5+7

7+5

6+6

12

it can be expressed in 89 ways