Suppose you have sufficient amount of rupee currency in three denominations: `₹1, ₹ 10 and `₹50. In how many different ways can you pay a bill of  ₹107?

The correct option is 18.

We are asked to find the number of different ways to pay a bill of ₹107 using three denominations of currency: ₹1, ₹10, and ₹50. Let us denote:
Let $x$ be the number of ₹50 notes/coins used.
Let $y$ be the number of ₹10 notes/coins used.
Let $z$ be the number of ₹1 notes/coins used.
Since the number of coins/notes must be non-negative integers, we have $x,y,z\ge 0$.

The total bill amount is ₹107, which gives us the linear equation:
$50x+10y+z=107$

Since we have a sufficient amount of each currency, we can choose non-negative integer values for $x$ and $y$ such that their combined value does not exceed 107. Once $x$ and $y$ are chosen, the value of $z$ is uniquely determined as:
$z=107-(50x+10y)$

Let us analyze the possible values for $x$ (the number of ₹50 notes):
Since $50x\le 107$, the possible non-negative integer values for $x$ are $0$, $1$, and $2$.

Case 1: $x=2$
Substituting $x=2$ into the equation:
$50\left(2\right)+10y+z=107$
$100+10y+z=107$
$10y+z=7$
For $y$ to be a non-negative integer, we must have $10y\le 7$, which means the only possible value is $y=0$.
This gives $1$ way: $(x=2,y=0,z=7)$.

Case 2: $x=1$
Substituting $x=1$ into the equation:
$50\left(1\right)+10y+z=107$
$10y+z=57$
For $y$ to be a non-negative integer, we must have $10y\le 57$.
The possible values for $y$ are $0,1,2,3,4,5$.
This gives $6$ ways.

Case 3: $x=0$
Substituting $x=0$ into the equation:
$10y+z=107$
For $y$ to be a non-negative integer, we must have $10y\le 107$.
The possible values for $y$ are $0,1,2,3,4,5,6,7,8,9,10$.
This gives $11$ ways.

To find the total number of ways, we sum the number of ways from all three cases:
$\text{Total ways}=1+6+11=18$

Thus, there are exactly 18 different ways to pay the bill.