R pays ₹100 to P with ₹5, ₹2 and ₹1 coins. The total number of coins used for paying are 40. What is the number of coins of denomination ₹5 in the payment?

The correct option is 13.

Let us solve the problem step-by-step by setting up a system of linear equations.

Let:
- $x$ be the number of ₹5 coins,
- $y$ be the number of ₹2 coins, and
- $z$ be the number of ₹1 coins.

We are given two main conditions in the problem:

1. The total number of coins is 40:
$x+y+z=40$ (Equation 1)

2. The total value of the coins is ₹100:
$5x+2y+1z=100$ (Equation 2)

We want to find the number of ₹5 coins, which is $x$. From Equation 1, we can express $z$ in terms of $x$ and $y$:
$z=40-x-y$

Now, substitute this expression for $z$ into Equation 2:
$5x+2y+(40-x-y)=100$

Simplify the equation:
$4x+y+40=100$
$4x+y=60$
$y=60-4x$ (Equation 3)

Since the number of coins of any denomination must be a non-negative integer, we test the options provided for $x$ (number of ₹5 coins) to see which one yields valid positive integer values for both $y$ and $z$:

Let us test the correct option, $x=13$:
Substitute $x=13$ into Equation 3:
$y=60-4\left(13\right)=60-52=8$

Now, substitute $x=13$ and $y=8$ back into Equation 1 to find $z$:
$13+8+z=40$
$21+z=40$
$z=19$

Since $x=13$, $y=8$, and $z=19$ are all positive integers and satisfy all constraints, the number of ₹5 coins is indeed 13.