Final answer:
When two new toppings are added to a selection of 12 toppings, there are 144 more ways for Ian to choose three toppings.
Step-by-step explanation:
When choosing 3 toppings from the original selection of 12, there are 12 choose 3 possible ways to choose. This can be calculated using the combination formula: C(n, r) = n! / (r!(n-r)!), where n is the number of items to choose from and r is the number of items to choose. In this case, n = 12 and r = 3. So the number of ways to choose 3 toppings from the original selection of 12 is:
C(12, 3) = 12! / (3!(12-3)!) = 12! / (3!9!) = (12 x 11 x 10) / (3 x 2 x 1) = 220
If two new toppings are added, the new selection of toppings would have 14 choices. To calculate the number of ways to choose 3 toppings from the new selection, we can again use the combination formula:
C(14, 3) = 14! / (3!(14-3)!) = (14 x 13 x 12) / (3 x 2 x 1) = 364
Therefore, the number of additional ways that Ian can choose three toppings after two new toppings are added is: 364 - 220 = 144.