Question

Pasta comes with tomato sauce and can be ordered with​ some, all, or none of these ingredients in the​ sauce: {onions,​ garlic, carrots,​ broccoli, shrimp,​ mushrooms, zucchini, green​ pepper}. How many different variations are available for ordering pasta with tomato​ sauce?

Answer 1

Answer: Hello!

So the list of ingredients in the sauce are:

{onions,​ garlic, carrots,​ broccoli, shrimp,​ mushrooms, zucchini, green​ pepper}

there are 8 possible elements in the sauce, so we need to see the combinations for this elements in the sauce.

let's count!

0 ingredients; there is only one combination with 0 ingredients.

1 ingredient: there are 8 combinations with 1 ingredient, one for each.

2 ingredients: now star playing with combinatorics, so the combinatory between A and B is: $\frac{A!}{B!(A-B)!}$, so for 8 and 2 we get:

$\frac{8!}{2!(6)!} = 8*7/2 = 28$

3 ingredients: Similar as before; the combinatory for 8 and 3 is: $\frac{8!}{3!*5!} = 8*7 =56$

4 ingredients: for 8 and 4 we have: $\frac{8!}{4!*4!} = \frac{8*7*6*5}{4*3*2} = 2*7*5 = 70$

5 ingredients: You can think that now we are counting the combinations of 3 ingredients that we are not using, so the combinations are the same as for 3 ingredients: 56

6 ingredients: Similar as before, here are the combinations of the two ingredients that we are not using: 28

7 ingredients: there are 8 options if we decide to not use only one ingredient.

8 ingredient: there is 1 option in this case, same as the case with no ingredients

And now, we need to add all those combinations:

so C = 1 +8 +  28 + 56+ 70+ 55 + 28 +8 +1 = 512 total combinations.