Thread: how to find number of different combination of 100

How many different ways can one hundred be written as a sum of at least two positive integers?

I am not getting algo. please any one can give me write solution of this problem

Originally Posted by freakycoder

...I am not getting algo. please any one can give me write solution of this problem

That's not how this forum works.

Originally Posted by freakycoder

How many different ways can one hundred be written as a sum of at least two positive integers?

I am not getting algo

Does "algo" mean algorithm?

Well...

This is related to a well-known topic in number theory: Integer Partitions

Here is an illustration of how calculation of a partition number (size of the set of partitions of a positive integer) relates to your assignment:

Let's look at the number of ways of writing a particular number, 5, as the sum of positive integers:

Note that p(5) = 7 and the seven ways of writing 5 as the sum of positive integers are:

5
4 + 1
3 + 2
3 + 1 + 1
2 + 2 + 1
2 + 1 + 1 + 1
1 + 1 + 1 + 1 + 1

Note also that, except for the first one, each of these lines (called parts or partitions of the number) consists of the sum of two or more positive integers.

Making a slight inductive leap (without formal proof):

The algorithm for finding the number of ways of writing 100 as the sum of two or more positive integers is:

1. Calculate p(100).
   
   
2. The answer is p(100)-1.

Finally, note that using partition number calculations as outlined in the Wikipedia article (and myriads of other references) depends on the following convention:

Two sums that differ only in the order of their terms are considered to be the same partition.
That is, 2+3 is the same as 3+2, so you don't count that as two different partitions.
Similarly, 2+2+1 is the same as 2+1+2, and is the same as 1+2+2.
Etc.

Cheers!

Z