Hi. I am trying to solve this problem.

For convenience, I have summarised the problem statement below:

**Problem Statement**

We are tasked to arrange *X* Apples, *Y* Cherries and *Z* mangoes on a line, Compute the number of valid arrangements (modulo 1*e*9 + 7). All *X* + *Y* + *Z* fruits must be used in a single arrangement.

A valid arrangement is defined as an arrangement in which no two fruits of a similar kind are adjacent (no wraparound).

Constraints: 1 ≤ *X*, *Y*, *Z* ≤ 2*e*5

Time limit: 2s

**Example**

Example of a valid arrangement for *X* = 2, *Y* = 2, *Z* = 2 is:

*Apple Orange Cherry Apple Orange Cherry*

Example of a invalid arrangement for *X* = 2, *Y* = 2, *Z* = 2 is:

*Apple Apple Orange Cherry Orange Cherry*

**My analysis**

If the constraints were smaller, an obvious dynamic programming solution would suffice. However, the large constraints here seem to point towards a combinatorics solution (which I cannot figure out).

Could someone please advise me on how to solve this problem?