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*MATH TRICK-13*

**Theorem:1:- The number of ways to divide n identical things among r persons when each get**

**at least one is (n-1)C(r-1) .**

__Example:__10 identical balls can be put in 4 different boxes in a row such that no box remains empty, is (10-1)C(4-1) =9C3 .

**RESULT-1:- The number of solutions of the equation x+y+z+ . . . . +k = n for positive integers**

**( each x,y,z, . . .,k > 0 ) with total number of element=r, is**

**(n-1)C(r-1) .**

**Theorem:2:- The total number of ways of dividing n identical things among r persons, each one of whom can receive 0,1,2,3 or more things (≤ n) is (n+r-1)C(r-1) .**

__Example:__30 mangoes can be distributed in 5 boys in (30+5-1)C(5-1) ways.

**RESULT-2:-**

**The number of solutions of the equation x+y+z+ . . . . +k = n for non-negative integers (**

**each x,y,z, . . .,k**≥

**0 ) with total number of element=r, is**

**(n+r-1)C(r-1)**

**.**

**PROOF : See Any Standard Book.**

**Problem: How many non-negative integer solutions are there of the equation x+y+z=18.**

__Sol:-__Use the Result-2 given above.

Here n=18, r=3 . So the answer is = 20C2.