GMAT Prep Combo Question

[sudaif]

I get the fact that we can assemble 8 different types of packages that contain 3 notepads of same size and color.
What I DON'T get is the latter number of arrangements.
This is how I approached it:
The 3 packages must have same size but (3) different colors.
First package: 2 sizes * 4 colors
2nd package: 1 size * 3 colors -----> there is only one option for size thus, number of choices for size = 1
3rd package: 1 size * 2 colors
Putting all that together, I get 48 different ways of having a package that contains notepads with the aforementioned specifics.
Thus in total, there would be 8 + 48 = 56 arrangements possible.

All solutions on forums involve the Combination formula. I have not bothered to re-learn the combinations formula -- since I've been advised to use either the slot method or the anagram method to solve such questions. Instructors, is it necessary to learn the Combo and Perm formulas and their application? Will appreciate help on this question!

A certain office supply store stocks 2 sizes of self-stick notepads, each in 4 colors: Blue, Green, Yellow Or Pink. The store packs the notepads in pacakages that contain either 3 notepads of the same size and the same color or 3 notepads of the same size and of 3 different colors. If the order in which teh colors are packed is not considered, how many different packages of the types described above are possible?

A) 6
B) 8
C) 16
D) 24
E) 32

[adiagr]

I get the fact that we can assemble 8 different types of packages that contain 3 notepads of same size and color.
What I DON'T get is the latter number of arrangements.
This is how I approached it:
The 3 packages must have same size but (3) different colors.
First package: 2 sizes * 4 colors
2nd package: 1 size * 3 colors -----> there is only one option for size thus, number of choices for size = 1
3rd package: 1 size * 2 colors
Putting all that together, I get 48 different ways of having a package that contains notepads with the aforementioned specifics.
Thus in total, there would be 8 + 48 = 56 arrangements possible.

All solutions on forums involve the Combination formula. I have not bothered to re-learn the combinations formula -- since I've been advised to use either the slot method or the anagram method to solve such questions. Instructors, is it necessary to learn the Combo and Perm formulas and their application? Will appreciate help on this question!

A certain office supply store stocks 2 sizes of self-stick notepads, each in 4 colors: Blue, Green, Yellow Or Pink. The store packs the notepads in pacakages that contain either 3 notepads of the same size and the same color or 3 notepads of the same size and of 3 different colors. If the order in which teh colors are packed is not considered, how many different packages of the types described above are possible?

A) 6
B) 8
C) 16
D) 24
E) 32

Hi,

Part (1): Pacakages that contain 3 notepads of the same size and the same color:

size 1: 4 ways (There are 4 colors)
size 2: 4 ways (-ditto-)

Total 8 ways.


Part 2: 3 notepads of the same size and of 3 different colors

Use slot method


Size 1

3 slots are there.

slot 1: can be filled in 4 ways

slot 2: can be filled in 3 ways

slot 3: can be filled in 2 ways

total ways: 4x3x2: 24

BGY is same as GYB so divide by 3!.

(24/3!) = (24/6) = 4 ways

Size 2

Similar slot method

4 ways

So Part (2) total ways: 8

Total ways : Part (1) + Part (2) = 16.

Aditya

[sudaif]

nice one aditya. forgot that the BGY is the same as BYG.
where did you learn your combo & perm theory? as in which book or you just knew it from before?
thanks

[Viswanathan.harsha]

adiagr, thats an interesting method on how you solved this problem. Can you please explain to me why you divided by 3! ? I am trying to understand the theory behind that and when to apply that.

[adiagr]

adiagr, thats an interesting method on how you solved this problem. Can you please explain to me why you divided by 3! ? I am trying to understand the theory behind that and when to apply that.

Guys That is "Slot method" which I learnt from this site. Search for permutation combination problems ....Ron and others have frequently discussed "Slot method".

[sudaif]

adiagr, thats an interesting method on how you solved this problem. Can you please explain to me why you divided by 3! ? I am trying to understand the theory behind that and when to apply that.

adiagr - the important thing here to realize is that Order does NOT matter. thus, the number of arrangements is reduced by 3!
Maybe an example will help.
Say there are three kites, of three different colors. Red - Blue - White (yes, its an american kite!)
now, if I were to ask you that, how many different ways can you fly the kites today? The first question you ought to ask me is, does the ordering of the color matter?
If the ordering does matter, then it can be said that there are 3 * 2 * 1 = 6 ways of flying the kites. You can also list these out...
On the other hand, if ordering does NOT matter, then essentially, R - B - W is the same as R - W - B or W - R - B or any of the other possible arrangements. Thus, essentially there is only one possible ordering of the kites -- which can be calculated by taking (3 * 2 * 1) / 3! = 1 way.

[RonPurewal]

adiagr, thats an interesting method on how you solved this problem. Can you please explain to me why you divided by 3! ? I am trying to understand the theory behind that and when to apply that.

for about an hour's worth of treatment on this topic, watch the DECEMBER 3 recording at the following link:

http://www.manhattangmat.com/thursdays-with-ron.cfm

[RonPurewal]

by the way, guys, don't forget one powerful but simple technique for combinatorics:
if you're working a combinatorics problem on which there is a SMALL NUMBER OF POSSIBILITIES:
if you don't IMMEDIATELY know how to calculate the answer, JUST START LISTING THE POSSIBILITIES.

if you look at the answer choices here, you'll quickly notice that they are all SMALL NUMBERS.
all of them are quite easily small enough to list in two minutes -- one minute, even. note that the maximum number of possibilities that you'll have to list is 25, since anything greater than 24 would mean (e).

--

let's make a list.

let's call the colors 1, 2, 3, and 4.

SMALL SIZE
SAME COLOR:
1, 1, 1
2, 2, 2
3, 3, 3
4, 4, 4
3 DIFFERENT COLORS:
1, 2, 3
1, 2, 4
1, 3, 4
2, 3, 4
that's it.

LARGE SIZE
exactly the same lists as for the small size.

therefore:
8 possibilities for the small size (just count them)
8 possibilities for the large size (same possibilities!)
----
16 possibilities total.

------------------------

i'll note that it took me less than two minutes to type this entire post -- including all of the explanatory text -- so you can definitely do this work in way, way, way less than the allotted time for this problem.

[psnvchaitanya]

Part (1): Pacakages that contain 3 notepads of the same size and the same color:

size 1: 4 ways (There are 4 colors)
size 2: 4 ways (-ditto-)

Total 8 ways.

Hi,

The second part is absolutely fine.

Can u please explain this part1.

My doubt is since there is only 1 notepad with same size and same color how do u pack them. To pack you require 3 notepads of same color and same size.

[RonPurewal]

My doubt is since there is only 1 notepad with same size and same color how do u pack them. To pack you require 3 notepads of same color and same size.

the problem does not state that the stock of notepads is limited, so there's no reason to assume that.