How many types of pizza combinations are possible? Before I begin, this question is from Algebra Nation's ACT prep subject, section 10: Statistics and Probability, and topic 5: Counting techniques.
It is given that:
Anthony's pizza offers 4 different types of cheese, 3 different kinds of meat toppings, 6 different kinds of vegetable toppings, and 2 different types of sauce. Each type of pizza on the menu has a combination of exactly 4 ingredients: 1 cheese, 1 meat, 1 vegetable, and 1 sauce. How many types of pizzas are possible?
The answer choices are as given:
A. 15
B. 63
C. 120
D. 144
E. 720
What I Tried
I've never really come across a problem like this, but I used my knowledge to try and come up with an answer.
I started by adding the different cheeses, meats, vegetables, and sauce types.
4+3+6+2=15
Since there are 4 ingredients I divided 4 by 15, but I immediately knew this would leave me with an answer that is not valid. Dividing 4 by 15 would give me a decimal, but I continued to go further into solving the problem.
⁴⁄₁₅=0.26
This answer was not applicable, so I tried multiplying 0.26 by 15.
0.26×15= 3.9
3.9 was nowhere near the values of the answer choices, which concludes that my method of solving the problem was not applicable.
I am unsure where I made my mistake exactly. But there is a chance that the whole problem was solved incorrectly and that neither of the steps was performed accurately.
 A: This is known as the "rule of product". Namely, if you have two lists $$1,\dots, n$$ $$1,\dots, m$$ Where one list is of length $n$ and the other list is of length $m$. Then the number of ways to pick one element from the first list and one element from the second list is $m\times n$. This can easily be generalised to when you have any number of lists and not just two lists. In your question, there are a total of $4$ lists. So the number of ways to choose $4$ elements where each comes from one list would be $$4\cdot 3\cdot 6\cdot 2=144$$
I think it would help to understand the logic behind counting how many ways there are to pick two things from two lists of things. Then you can generalise this reasoning to any number of lists.
Consider the two lists of numbers $$1, 2, 3\\ 4, 5, 6$$
How many ways can I pick one number from the first list and one number from the second list?
Well, if we fix one number $x$ from the first list. Then there are three choices for us to pick from the second list. This tells us that there are three ways of picking a number from the second list given a number $x$ from the first list. Thus every number $x$ from the first list gives rise to three combinations. Because there are $3$ numbers in the first list and each of them would give us three combinations. The total number of combinations is $$3+3+3=3\times 3=9$$
As an exercise: Try this with a two lists that have $4$ and $5$ elements respectively. Then try this with $3$ lists of any length of your choice. Use the result from $2$ lists to help you with finding how many ways there are to pick elements from $3$ lists.
Edit: A useful restatement of the rule of product/principle of multiplication(that I just remembered) is as follows.
Statement: If an event $E$ can be broken down into $k$ events $E_1,\dots,E_k$ such that there are $$n_1\text{ ways of doing }E_1 \\ n_2\text{ ways of doing }E_2\\ \vdots \\ n_k\text{ ways of doing }E_k$$ Then there are a total of $n_1n_2\cdots n_k$ ways of doing $E$.
In your question the $E$ can be replaced by "how many pizzas can we make". This even can be broken down into $4$ events $$E_1=\text{ number of cheese}\\ E_2= \text{ number of meat}\\ E_3=\text{ number of vegetables }\\ E_4=\text{ number of sauces}$$
Then the multiplication principle tells us that the number of ways $E(\text{how many pizzas can we make})$ can happen is $$4\cdot 3\cdot 6\cdot 2=144$$
A: This is a basic counting question. This question might seem very complicated at the beginning but is just need some sparks of light.
First of all, this question belongs to the beautiful branch of mathematics called "Combinatorics" particularly Enumerative Combinatorics. The answer to this question comes from a very intuitive reasoning, namely

If I wanna count how many ways are there to combine two or more objects of different kind, I may multiply the number of objects (chooses) of any kind altogether

This is the principle of multiplication. Although this may seem rather contrived, it isn't if you take an example.
If I'd have $7$ different shirts and $8$ different trousers in how many ways can I wear a different outfit?
We should poke and prod the matter. The key indeed is to see that for every shirt there are $8$ ways to choose a trouser to wear together and since there are $7$ different shirts the answer is then $8 \cdot 7 = 56$.
Your problem is very similar because to make a pizza means choose a type of cheese, meat, vegetable and sauce thus the number of different pizzas is the same as the number of ways of combine these ingredients or in other words
$$ 4 \cdot 3 \cdot 2 \cdot 6 = 144$$
There are many good books, videos, articles, problems etc. To introduce yourself this topic of combinatorics but I found Chapter 6.1 of The Art and Craft of Problem Solving by Paul Zeitz one of the most easygoing and understandable of them all. Check it out if you are interested in learning more fun facts!
A: It's a basic problem. You have 4 options to choose cheese, 3 options to choose meat, 6 options to choose vegetable, and 2 options to choose sauce.
The answer $= 4 * 3 * 6 * 2 = 144.$
