Question

In a bakery four types of biscuits are available. In how many ways a person can buy $10,$ biscuits if he decide to take at/east one biscuit of each variety$?$

Answer 1

Let the person select $x$ biscuits from first variety, $y$ from the second, $Z$ from the third and $w$ from the fourth variety. Then the number of ways=number of solutions of the equation $x+y+z+w=10.$
where $x=1,2,...,7$
$y=1,2,...,7$
$z=1,2,...,7$
$w=1,2,...,7$
So, number of ways=coefficient of $x^{10}$ in ${(x+x^2+...+x^7)}^4$
=coefficient of $x^6$ in ${(1+x+...+x^6)}^4$
=coefficient of $x^6$ in ${(1-x^7)}^4{(1-x)}^{-4}$
=coefficient $x^6$ in ${(1-x)}^{-4}$
${=}^{4+6-1}{\text{C}}_6{=}^9{\text{C}}_3=84$