Question

Suppose you go to an ice cream shop which allows you to choose from 7 flavors of ice cream and 5 toppings. if you decide to get a sundae with 2 different types of ice cream and 3 different toppings, how many ways are there to make this choice?

Answer 1

The order doesn't matter.

$k$ objects can be chosen out of $n$ objects, when the order doesn't matter, in $C(n,k)=\dfrac{n!}{k!(n-k)!}$ ways.

So, the answer is $C(7,2)\cdot C(5,3)=\dfrac{7!}{2!5!}\cdot\dfrac{5!}{3!2!}=\dfrac{6\cdot7}{2}\cdot\dfrac{4\cdot5}{2}=210$ ways.