How many combinations of meals at a Sub resturant?

Question

There are $6$ different breads, $16$ different meats, $5$ different cheeses, $12$ different vegetables, $16$ different sauces, $5$ different chips, $5$ different cookies, $18$ different drinks. The bread can either be 6 inch or 12 inch, toasted or not toasted.

a) Calculate the number of possible subs.

I thought of $83!$ (total number of elements)divided by $5!5!5!18!16!16!12!6!$, not sure this will get me the right answer though.

b) How would the number of possible subs change if we include double meat and/or double cheese into our possibilities?

c) How many options are possible for a chicken sub with lettuce (everything else is possible)

d) If you don’t like peppers, how many options are possible?

Answer 1

Assuming a meal consists of one selection from each category you listed there would be $6\cdot16\cdot5\cdot12\cdot16\cdot5\cdot5\cdot18$ total meals you could create.

It may help you to read up on something called the "rule of product": https://brilliant.org/wiki/rule-of-product/

Answer 2

The answer is $$ 5\times 5\times 5\times 18\times 16\times 16\times 12\times 6 $$ It is a direct application of the multiplication principle

P.S.: your formula of $$ \frac{83!}{ 5!5!5!18!16!16!12!6!,} $$ computes the total number of "Permutations with Repetition of Indistinguishable Objects" that is, the number of $83$-tuples that can be formed with $6$ identical pieces of bread, $16$ identical pieces of meat, $5$ identical pieces of cheese, $12$ identical pieces of vegetable, $16$ identical sauces, $5$ identical chips, $5$ identical cookies, $18$ identical drinks.

Answer 3

The cookies and the the drink are not part of the sub, so they should be excluded from the "number of possible subs" calculation.

The follow on question discusses double meat and double cheese. So one meat / cheese seems to be assumed.

And no meat or no cheese are valid option.

There is definitely only one type of bread.

You can put mustard and mayo, on a sandwich, while I don't know if I want 16 different sauces on my sandwich, is that a possibility? or are we going with one sauce?

What about mustard and mayo? Is that the only one we double up?

I am going to go with $1$ sauce (or dry)

The veggies: Any or all may be on a given sandwich.

So, there are $12$ varieties of veg. Each one could be on or could be off the sandwich that is $2^{12}$ veg options.

My best guess, due to the unclear constraints would be

$6\cdot 17\cdot 6\cdot 17\cdot 2^{12} $

Double meat double cheese, and including all of the single meat, single cheese, all veg options

$6\cdot (17+\frac {16\cdot 15}{2})\cdot (6+\frac {5\cdot 4}{2})\cdot 17\cdot 2^{12} $

c) the meat is chicken, and we are definitely getting the lettuce. Is double cheese still an option?

$6\cdot (6+\frac {5\cdot 4}{2})\cdot 17\cdot 2^{11} $

d) no peppers

$6\cdot (17+\frac {16\cdot 15}{2})\cdot (6+\frac {5\cdot 4}{2})\cdot 17\cdot 2^{11} $