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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: , so for 8 and 2 we get:
3 ingredients: Similar as before; the combinatory for 8 and 3 is:
4 ingredients: for 8 and 4 we have:
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.