Question

At the grocery store, marlon has narrowed down his selections to 44 vegetables, 55 fruits, 66 cheeses, and 77 whole grain breads. he wants to use the express lane, so he can only buy 2020 items. in how many ways can he choose which 2020 items to buy if he wants all 77 whole grain breads?

Answer 1

105 if the vegetables, fruits, and cheeses are unique.
6 if the vegetables, fruits, and cheeses are interchangeable.

There are 2 possible solutions to this problem depending upon how you actually look at the problem. Both solutions assume that the problem gets simplified to "What 13 items out of the 4 vegetables, 5 fruits, and 6 cheeses, will Marlon select?" The reason is because Marlon is going to select the 7 breads, so he can only select 13 more items.

First possible solution is that the vegetables, fruits, and cheeses are unique. For instance he has peas, carrots, tomatoes, and beans as his vegetables and the same for fruits and cheeses. For that situation, the number of ways to fill the remaining 13 slots is 15!/(13!2!) = 105 possible ways.

The second possible solution is that he has 4 identical vegetables, 5 identical fruits, and 5 identical cheeses. Since each item of each type is identical, there isn't any real distinction between them. So let's simplify the problem and instead as "Which 2 items Marlon won't buy?" and the answer to that question will be the same as the number of things he will buy. But it will be small enough to completely enumerate the possible solutions. Marlon may throw away as the first item any of a vegetable, fruit, or cheese for a total of 3 possibilities. And for the 2nd item, it can also be any of a vegetable, fruit, or cheese. So you have 9 possibilities. But that's not completely true. There's no real difference between vegetable/cheese and cheese/vegetable. So if you ignore the order, there's a total of 6 different combinations that Marlon can toss out of his cart. And therefore there is a total of 6 possibilities for Marlon to fill the remaining 13 slots he has in order to use the express lane.