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I need to figure out the total number of combinations for three categories of items which must be combined, and I'm trying to determine the appropriate way to do so. Let's say the categories are as follows:

Shirts: 3 Total, must choose 1 Ties: 18 Total, must choose 1 Additional Accessories: 20 Total, must choose 4

I must choose 1 shirt, 1 tie, and 4 accessories, which constitutes an outfit.

I understand how to develop a "normal" combinatorics problem (have 20, choose 4), but I'm not sure I understand the appropriate way to develop the total number of "outfits" from this example...What is the appropriate way to determine how many outfits can be created? Thank you in advance,

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  • $\begingroup$ You have 3 choices for the shirt, 18 choices for the tie, and $\binom{20}{4}$ possible combinations of accessories. So by the product rule, what is your total? $\endgroup$ – Mauve Jan 24 '18 at 21:31
  • $\begingroup$ Well, I think, by rule of product it would be 261,630...That's multiplying through the combinations for each. Is that correct? I'm trying to determine if that's #1, appropriate, and #2, correct...Is this the correct application of Rule of Product? $\endgroup$ – jules325 Jan 24 '18 at 21:33
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The answer is $$\binom{3}{1}\binom{18}{1}\binom{20}{4} = 261630$$ because when we make an outfit by choosing a shirt, a tie and four accessories seperately, each and every choice will be different from each other. That's why we can directly apply product rule as Useful (nevermind his current name :D) stated.

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    $\begingroup$ Thank you @ArsenBerk, @ Useless, I thought that was the way to go about it but I'm not super comfortable with combinatorics, and also the sheer size of the output was throwing me off...I mean that just seems enormous... $\endgroup$ – jules325 Jan 24 '18 at 21:41

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