A pizza shop offers 7 different toppings. If a special pizza must have exactly 3 distinct toppings chosen from these 7 and the order of toppings on the pizza does not matter, how many different 3-topping pizzas can be made?

Introduction / Context:
This problem involves combinations in a simple and familiar context: choosing pizza toppings. We are selecting topping combinations where placement order on the pizza does not matter, only which toppings are included. This is a very typical example used to illustrate combinations in many introductory probability and combinatorics courses.

Given Data / Assumptions:

• Total types of toppings available: 7.
• Each topping is distinct.
• We must choose exactly 3 different toppings for a special pizza.
• The same topping is not chosen more than once.
• The order in which toppings are chosen or appear on the pizza does not matter.

Concept / Approach:
Since we are selecting a set of toppings and only the combination matters, this is a pure combination question. The number of ways to choose r toppings from n available toppings is given by:
nCr = n! / (r! * (n - r)!). Here, n = 7 and r = 3, so we must compute 7C3 to get the number of different 3 topping pizzas.

Step-by-Step Solution:
Step 1: Identify n = 7 and r = 3. Step 2: Use the combination formula: 7C3 = 7! / (3! * 4!). Step 3: Simplify 7! / 4! to 7 * 6 * 5. Step 4: Compute numerator: 7 * 6 * 5 = 210. Step 5: Denominator is 3! = 3 * 2 * 1 = 6. Step 6: Divide 210 by 6 to get 35. Step 7: Therefore, there are 35 distinct 3 topping pizza combinations.

Verification / Alternative check:
We can also reason that once you pick any 3 toppings, there is exactly one pizza corresponding to that set, since order and layout are not considered different. Checking with symmetry, 7C3 equals 7C4 as well, because choosing 3 toppings to include is equivalent to choosing 4 to exclude. Computing 7C4 leads to the same result, confirming 35 as correct.

Why Other Options Are Wrong:
49: Equal to 7^2, which would correspond to ordered pairs, not unordered triples. 27: Equal to 3^3, unrelated to choosing 3 items from 7. 25: A random smaller number that does not come from the combination formula. Only 35 correctly represents 7C3.

Common Pitfalls:
Some students mistakenly think that order matters and use permutations, leading to a much larger number than needed. Others might misinterpret the question and allow repeated toppings, which is not intended when we say distinct toppings. Remember that for combinations such as toppings, flavors, or colors, unless explicitly ordered, we use nCr, not nPr.

Final Answer:
The number of different 3 topping pizzas that can be made from 7 toppings is 35.