I have 15 different ties (7 blue, 3 red and 5 green), 4 waistcoats (red, black, blue and brown) and 12 different shirts (3 each of red, pink, white and blue). I always wear a shirt, tie and waistcoat, but will never wear an outfit that combines red with pink.
How many different outfits can I wear?
Combinatorics is not a topic that I am comfortable with, and would appreciate how someone would go about this. Ideally, I am looking for a method that does not involve combinations and permutations directly (just using the product and addition principle). Thanks.
Best Answer
Since the only pink things are your shirts, I would split this into two cases:
All possible outfits with a pink shirt
All possible outfits with a non-pink shirt
In the first case, after picking one of your $3$ pink shirts, you can no longer choose a red tie, and so you are left with $12$ blue or green ties. Likewise, you cannot wear the red waistcoat, and so are left with $3$ options there. This gives a total of $3 \cdot 12 \cdot 3 = 108$ possible outfits with a pink shirt.
In the second case, you start with any of your $9$ non-pink shirts, after which you can pick any of your ties ($15$ options), any of your waistcoats ($4$ options), for a total of $9 \cdot 15 \cdot 4 = 540$ possible outfits with a non-pink shirt.
Total: $108 + 540 = 648$ possible outfits