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

There is no direct formula for this

a) if 6 integers

all are two

1 way

b) if 5 integers , 3 two\'s 2 three\'s OR 4 two\'s one four

no of ways 5C2 + 5C1

there is this direct formula

No of ways = integer solutions for the equation

x1 + x2 + x3+ x4+ x5 = 12 where x1,x2,x3,x4,x5 >=2

which is same as

integer solutions for the equation

x1 + x2 + x3+ x4+ x5 = 2 where x1,x2,x3,x4,x5 >=0

which is equal to 6 Choose 2 = 15

c) four integers

integer solutions for the equation

x1 + x2 + x3+ x4 = 4 where x1,x2,x3,x4 >=0

which is equal to 7 Choose 4 = 35

d) three integers

integer solutions for the equation

x1 + x2 + x3 = 6 where x1,x2,x3 >=0

which is equal to 8 Choose 2 = 28

d) three integers

integer solutions for the equation

x1 + x2 = 8 where x1,x2 >=0

which is equal to 9 Choose 1 = 9

e) 1 integer

no of ways = 1

So total ways = 89