2. Given a positive integer

define a

-partition to be a sum of

positive integers which sum to

. For example,

. The following are

-partitions.

and

and

. Notice that

are considered distinct*. Say you a given a specific

. And given a specific value of

, can you find the total number of

-partitions of this integer, with a formula?** Now try to see how many partitions (again not counting order) exist for a given integer

(the answer is really supprising).

Hint: Review your Combinatorics formula for this one.