A Summary of the Four Selection Schemes and Combinatorics in Salad Making with 3 Ingredients

The table below gives a summary of the four selection schemes.

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Question)

We have an unlimited supply of tomatoes, bell peppers and lettuce. We want to make a salad out of 4 units among these three ingredients (we do not have to use all ingredients). The order in which we use the ingredients does not matter. How many different salads we can make?

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Answer)

To solve this problem, we can use the concept of combinations with repetition, also known as "stars and bars" in combinatorics. We have three types of ingredients (tomatoes, bell peppers, and lettuce), and we want to choose a total of 4 units of these ingredients to make a salad. The order of ingredients doesn't matter, so we are essentially looking for the number of ways to distribute 4 identical units among 3 distinct categories.

The formula for combinations with repetition is given by:

where n is the number of categories (in our case, 3 ingredients) and k is the number of units to choose (4 units in our case). This formula calculates the number of ways to choose k items from n categories with repetition allowed.

Let's calculate the number of different salads we can make.

from itertools import combinations_with_replacement
for c in combinations_with_replacement("TBL", 4):    
print("".join(c))

TTTT
TTTB
TTTL
TTBB
TTBL
TTLL
TBBB
TBBL
TBLL
TLLL
BBBB
BBBL
BBLL
BLLL
LLLL

6!/4!2! = 15

There are 15 different salads that can be made using 4 units of tomatoes, bell peppers, and lettuce, with the combination of ingredients chosen from these three types.