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How To Find Number Of Integral Solutions Of Linear Equations

MATH TRICK-13

Before explaining the Trick let me explain some of the basic principles of Combinatorics.
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.