Three men have 4 coats, 5 waist coats, and 6 caps

Question

Three men have 4 coats, 5 waist coats, and 6 caps. In how many ways can they wear any one type of them?

Source: The combinatorics tutorial on gmatclub.com.

While I could find the solution, I'm having confusion understanding why another method does not work. This is the solution given by the author of the post:

Number of ways in which 3 Men can wear 4 coats = 4! = 24
Number of ways in which 3 Men can wear 5 waist coats = 5!/2! = 5 X 4 X 3 = 60
Number of ways in which 3 Men can wear 6 caps = 6!/3! = 6 X 5 X 4 = 120

By Fundamental Principle of Addition

Arrangement of 4 Coats OR Arrangement of 5 Waist Coats OR Arrangement of 6 Caps

Total number of ways = 24 + 60 + 120 = 204

My doubts:

1. Why can't I approach the problem as follows: 
Number of ways in which 1st man can be allocated a coat: 4
Number of ways in which 2nd man can be allocated a coat:3
and so on.

Thus, number of ways in which 4 coats can be allocated: 4X3X2 =12

Similarly, number of ways in which 5 waist coats can be allocated: 5X4X3=60
and 
number of ways in which 6 caps can be allocated: 6X5X4= 120

Thus, total number of ways =120+60+12=192

2. Why add the counts. Why not multiply them?
I usually don't have a problem differentiating between adding and multiplying in combinatorics problems, but this one is really confusing me.

THECEO123 · Answer

You have an error in your calculations -
4x3x2=24; you have 12

MBAHopeful99 · Answer

First, I have bolded an error you have made. This should be 24, bringing your total number of ways to 204.

For your second question, you must recognize that these scenarios are broken down into different outcomes. For example, the hat picked does not determine the coat picked. You are adding up these choice for the individual outcomes. Multiplying these scenarios would create outcomes that are not possible.

I hope that you understand.

VeritasPrepDennis · Answer

I agree with MBAHopeful99, but would add one thing: you would add the counts since they are independent events. If the events were dependent, then you would multiply.

samuraijack256 · Answer

Thanks, MBAHopeful99 and VeritasPrepDennis!

I understand now why I should be adding and not multiplying the decisions: They are independent of each other.

And the other obvious part: I definitely need to work on avoiding silly mistakes in my calculations

!

testtakerstrategy · Answer

This might be obvious, but why do we use permutations, rather than combinations here? How can we assume the "men" are distinct (ex. replace "men" with mannequin -- then we would get the same number, but divided by (3!). How do we know NOT to divide by 3! here?

bumpbot · Answer

Automated notice from GMAT Club BumpBot: 

A member just gave Kudos to this thread, showing it’s still useful. I’ve bumped it to the top so more people can benefit. Feel free to add your own questions or solutions.

 This post was generated automatically.

Three men have 4 coats, 5 waist coats, and 6 caps

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

GMAT Problem Solving (PS) Questions

Three men have 4 coats, 5 waist coats, and 6 caps

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards