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Dolly has 3 pairs of shoes, 4 sweaters (one red), 2 skirts (one orange), and 5 hats. Determine the number of possible outfits consisting of:

a. A sweater, a skirt, a pair of shoes, and a hat.
b. A sweater, a skirt, a pair of shoes, and a hat or no hat at all.
c. A sweater, a skirt, a pair of shoes, and a hat; but not the red sweater with the orange skirt.
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2 Answers

 
the number of possible outfits under each scenario is 
 
(sweater)(skirt)(shoes)(hats)
 
a. (4)(2)(3)(5)=120
b. In this case the number of choices of hats is 6 bc you can choose any of the 5 or none at all. 
So (4)(2)(3)(6)=144
 
c.  In this case you want the number of possible outfits 120 from part a  above and the subtract all the possible  outfits which red sweater w the orange shirt which is (3)(1)(1)(5) is 15 so answer is 120-15=105
 
a. 3x4x2x5=120
b. 120+(3x4x2)=144
c. 120-(3x5)=105
 
Note: Part b asks for the number of outfits without restriction OR the number of outfits without a hat;  thus we must give an answer that consist of the outfits with and without hats, hence the addition.
     Part c asks for an enumeration of the outfits with the restriction of omitting the combination of one skirt and one sweater; but realize that counting the amount of outfits that consist of this specific combination, and then subtracting this number from the total number of outfits will leave only the number of outfits without that one sweater and skirt combination.