Question

In how many different ways can $3$ persons $A,BandC$ having $6$ one rupee coins, $7$one rupee coin and $8$ one rupee coins respectively donate $10$ one rupee coins collectively.

Answer 1

Let $A,B,C$ donate $x_1,x_2,x_3$ coins with $x_i\ge 0$.

Then $A/Q\sum x_i=\mathrm{10......}\left(1\right)$

At first lets find all the solutions of this equation in integers.

The no. of such solutions is $12C2$

Now we will find out the no. of solutions in which $x_1\ge 7$, ( these solutions cant be considered),

To find this lets replace $x_1$ by $x+6$ where $x\ge 1$ putting this into $1$ we have $x+x_2+x_3=4$ we will find no. of such soln. $5C2$

In this way we will find the other cases which are not possible.

Namely when $x_2\ge 8$ in this case we have $4C2$ solutions, and the last one whne $x_3\ge 9$ then we have $3C_2$ solutions.

All the cases which cant be possible are disjoint implying the total no. of solution = ( total no. of cases)-(cases not possible).

$12C2-5C2-4C2-3C2$