Permutation & combinations problem- difficult one

[shailendra10]

Dave has no fashion sense, and will wear any combination of garments regardless of whether someone thinks they "match." Every day Dave chooses an outfit consisting of one of each of the following garments: jacket, tie, shirt, pants, boxers, right sock, left sock, right shoe, left shoe. If Dave has more than one of each of the listed garments, and can make 63,000 different outfits, then for how many garments does Dave have exactly five choices?

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A 0

B 1

C 2

D 3

E 4

I wish to understand the logic behind correct choice -D as shown below?
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Dave's garment choices are independent, so the number of each type of garment will be multiplied together to get 63,000. . Nine prime factors and nine types of garments, so there must be three garments of which Dave has exactly five choices.


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[manpsingh87]

Dave has no fashion sense, and will wear any combination of garments regardless of whether someone thinks they "match." Every day Dave chooses an outfit consisting of one of each of the following garments: jacket, tie, shirt, pants, boxers, right sock, left sock, right shoe, left shoe. If Dave has more than one of each of the listed garments, and can make 63,000 different outfits, then for how many garments does Dave have exactly five choices?

------------------------------------------------------------------
A 0

B 1

C 2

D 3

E 4

I wish to understand the logic behind correct choice -D as shown below?
--------------------------------------------------------------------------

Dave's garment choices are independent, so the number of each type of garment will be multiplied together to get 63,000. . Nine prime factors and nine types of garments, so there must be three garments of which Dave has exactly five choices.


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dave can make 63,000 different outfits, by arranging nine type of garments, also if we analyze the 63000, we will notice that it is made up of 7*3^2*5^3*2^3; i.e. for 3 garment dave has 2 choices;
for 3 (other than the previously selected) dave has 5 choices, for 2 garment it has 3 choices , and one garment it has 7 choices,

as seen therefore for 3 garments dave has exactly 5 choices hence D

[Frankenstein]

Hi,
As the events are independent, the number of ways is the product of the number of ways of all individual events.
Lets say a shirt has 2 choices, pant has 4 choices, tie has 5 choices, then the number of ways = 2.4.5
Here, in this question we employ similar logic.
63000 = (2^3)(3^2)(5^3).7=2.2.2.3.3.5.5.5.7
In the question there are 9 independent events and each has more than 1 way. So each independent event can happen in at least 2 ways and the product 63000 is made up of exactly 9 prime numbers and these 9 numbers are the number of ways of each of the 9 garments. It has three '5's. So, the number of garments having exactly 5 choices is 3.

Cheers!

[cans]

63000 => 63*1000 => 63*(5^3)*(2^3)
thus 3 types of clothes (because of 5^3)