Worked solution
Count nonempty subsets and a selection with tomatoes fixed
The salad bar at a restaurant has 6 separate bowls containing lettuce, tomatoes, cucumber, radishes, spring onions and beetroot respectively. John decides to visit the salad bar and make a selection. At each bowl, he can choose to take some of the contents or not.
(i) Assuming that John takes some of the contents from at least one bowl, find how many different selections he can make.
(ii) John decides he is going to have 4 salad items, and one of them will be tomatoes. How many different selections can he make?
[1999/I/18(b)]
CJC · Discussion · Question 11
Answer
(i) 63; (ii) 10.
(i) Method 1: binary choices minus the empty selection
For each bowl, take or do not take: possibilities. Exactly one takes nothing, so nonempty selections.
(i) Method 2: retain all six source cases
Selecting from 1, 2, 3, 4, 5 or 6 bowls gives , , , , and , respectively. Their disjoint sum is .
(ii) Include tomatoes, then choose three more
Tomatoes are already fixed. Choose 3 of the other 5 items: . Quantities of a chosen ingredient are not separate choices in the stated model.
Common mistakes
The empty selection is excluded in(i); in(ii), tomatoes use one of the four items.
Solution notes
All printed asks, working, counting factors and source annotations visually compared and retained. No marks are printed for this question.