Among 50 sandwiches made for the birthday party 28 had tomatoes, 17 had olives and 15 had onions. 10 had tomatoes and olives, 8 had tomatoes and onions, 4 had olives and onions. None of the sandwiches had all three options. Find the number of sandwiches with tomatoes or onions but without olives

To solve this problem, we need to find the number of sandwiches with tomatoes or onions but without olives.

Step 1: Find the total number of sandwiches with tomatoes or onions. This is the sum of the sandwiches with tomatoes and the sandwiches with onions. However, this will double count the sandwiches with both tomatoes and onions, so we need to subtract those out.

Total with tomatoes or onions = 28 (tomatoes) + 15 (onions) - 8 (both) = 35

Step 2: Find the total number of sandwiches with tomatoes or onions and olives. This is the sum of the sandwiches with tomatoes and olives and the sandwiches with onions and olives.

Total with tomatoes or onions and olives = 10 (tomatoes and olives) + 4 (onions and olives) = 14

Step 3: Subtract the total number of sandwiches with tomatoes or onions and olives from the total number of sandwiches with tomatoes or onions. This will give us the number of sandwiches with tomatoes or onions but without olives.

Total with tomatoes or onions but without olives = 35 (total with tomatoes or onions) - 14 (total with tomatoes or onions and olives) = 21

So, there are 21 sandwiches with tomatoes or onions but without olives.