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Combinatorics: The multiplication and addition principles
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Daniel and his friends are going to watch a movie. Daniel has 7 movies, his best friend has 5 movies, and his second best friend has 3 movies. How many different movies can they choose between?
Mikhael is going to a party and he is choosing clothes. He has 2 suits, 4 shirts, and 3 ties. How many combinations can he choose from? He can choose 1 of the 2 suits, so we have 2 possible choices, and he has 4 choices of the shirts, and on top of that, one of the 3 ties. Another 3 choices.
We can draw Mikhael's decision process as a tree like this. First, pick 1 of the suits. After that pick 1 of the shirts, and with a suit and a shirt chosen, pick 1 of the 3 ties. You get the total number of combinations by multiplying the number of choices at each step. 2 suits times 4 shirts times 3 ties.
This gives 24 possible combinations. No wonder it's hard to choose. Lena is also going to the party and she's choosing a dress. She can either take 1 of the 3 dresses in her wardrobe, or she can borrow 1 of the 5 dresses Maria said she could lend her, or she can buy 1 of the 4 dresses she saw in a second hand store down the street. How many choices does she have?
Now, we are not multiplying. We add. 3 in the wardrobe plus 5 at Maria's plus 4 in the store. Lena can choose from 12 dresses. There are 12 combinations.
Hold on. Why did we multiply for Mikhael, but add for Lena? A conspiracy? No. Mikhael had to choose a suit, a shirt, and a tie.
Lena, on the other hand, had to choose between dresses in the wardrobe, or at Maria's, or at the store. When we make several consecutive choices, the alternatives multiply. When we make choices between one group or another, the alternatives add. This is called the multiplication principle and the addition principle. If you're with us so far, let's proceed to a bit more complicated problems.
Mikhael has a choice not to wear a suit, but borrow Leon's tuxedo instead. Then he has his 24 previous choices in accordance with the multiplication principle, or Leon's tuxedo. This is 24 plus 1, or 25 choices. Well, that turned out well. The decisions have been made and the party can begin.