Question

In how many ways can $5$ pounds be paid in shillings and sixpences, including zero solutions( positive integer solutions )?

• A. $100$
• B. $101$
• C. $102$
• D. $103$

Answer 1

The correct option is B $101$

$1$ pound $=$ $20$ shillings

$1$ pound $=$ $40$ sixpences

Let the number of shillings be $x$ and the number of sixpences be $y$

According to question, we have

$\frac{x}{20}+\frac{y}{40}=5\Rightarrow \frac{2x+y}{40}=5\Rightarrow 2x+y=200$ .....(i)

We will approach the question by finding the number of positive integrals solution of (i) as the number of coins can only be integer.

$x+\frac{y}{2}=100$

As $x$ is an integer

$\Rightarrow \frac{y}{2}=$ integer

Let the integer be $p$

$\frac{y}{2}=p$

$\Rightarrow y=2p$ .....(ii)

Substituting $y$ in (i), we get

$2x+2p=200$

$\Rightarrow 2x=200-2p$

$\Rightarrow x=100-p$ ......(iii)

From (ii) we can see that $y<0$ for integer $p<0$ and from (iii) $x<0$ for integer $p>100$, which is not possible

So $p$ can take all integer values from $[0,100]$

So, the number of possible distributions is $101$.