Question

The total number of ways in which a beggar can be given at least one rupee from two $1$ rupee coins, three $50$ paisa coins and four $25$ paisa coins is

• A. $51$
• B. $52$
• C. $53$
• D. $54$

Answer 1

The correct option is D $54$

Let the beggar is given $x$ number of one rupee coins, $y$ number of $50$ paisa coins and $z$ number of $25$ paisa coins.

$\Rightarrow x+0.5y+0.25z\ge 1$
where $x=0,1,2;y=0,1,2,3;z=0,1,2,3,4$
$\Rightarrow 4x+2y+z\ge 4$

For $x=1,2,y$ and $z$ can take any value.
Number of ways $=2\times 4\times 5=40$

For $x=0\Rightarrow 2y+z\ge 4$

For $y=0,z=4\to 1\text{way}$
For $y=1,z=\{2,3,4\}\to 3\text{ways}$
For $y=2,z=\{0,1,2,3,4\}\to 5\text{ways}$
For $y=3,z=\{0,1,2,3,4\}\to 5\text{ways}$

Total number of ways $=40+1+3+5+5=54$