I am just wondering if I did these questions correctly as I do not have an answer key for them.

A second opinion would be greatly appreciated. Thanks.

A pizza company has 15 toppings to choose from, of which 6 are vegetables.

1. How many differentthree-toppingpizzas haveexactlyone vegetabletopping?

= (all possible combinations of 1 veggie topping) x (all possible combinations of 2 non-veggie toppings)

= $\displaystyle _{6} C _1$ x $\displaystyle _{9} C _2$

= 216

2. How many differentthree-toppingpizzas haveat leastone vegetabletopping?

= (all possible combinations of 3-topping pizzas) - (pizzas without any veggies)

= $\displaystyle _{15} C _3$ - $\displaystyle _{9} C _3$

= 371

3. How many differentthree-toppingpizzas haveexactlytwo vegetabletoppings?

= (all possible combinations of 2 veggie toppings) x (all possible combinations of 1 non-veggie topping)

= $\displaystyle _{6} C _2$ x $\displaystyle _{9} C _1$

= 135

4. How many differentthree-toppingpizzas haveat leasttwo vegetabletoppings?

= (all possible combinations of 3-topping pizzas) - (pizzas without any veggies) - (pizzas with exactly 1 veggie)

= $\displaystyle _{15} C _3$ - $\displaystyle _{9} C _3$ - ( $\displaystyle _{6} C _1$ x $\displaystyle _{9} C _2$ )

= 455 - 84 - 216

= 155

OR

= (pizzas with exactly 2 veggies) + (pizzas with exactly 3 toppings)

= ( $\displaystyle _{6} C _2$ x $\displaystyle _{9} C _1$ ) + $\displaystyle _{6} C _3$

= 135 + 20

= 155