Question:

Since notes have the same denomination, the HCF of 6, 7, and 8 is 1, so each note is $1. Let $x_1, x_2, x_3$ denote the number of each type of note. The number of ways to denote Rs. 10 is the coefficient of $x^{10}$ in $(1 + x + x^2 + \cdots + x^6)(1 + x + x^2 + \cdots + x^7)(1 + x + x^2 + \cdots + x^8)$. This equals the coefficient of $x^{10}$ in $\frac{(1-x^7)(1-x^8)(1-x^9)}{(1-x)^3}$, which simplifies to $^{12}C_2 - ^5C_1 - ^4C_2 - ^3C_1 = 47$.