GMAT PRep Combo question

[san2009]

A certain restaurant offers 6 kinds of cheese and 2 kinds of fruit for its desert platter. If each dessert platter contains an equal number of kinds of cheese and kinds of fruit, how many different dessert platters could the restaurant offer?

a. 8
b. 12
c. 15
d. 21
e. 27 is OA

My take - pls tell me how to correct my "SLOT" method approach
1 fruit 1 cheese
2 * 6 = 12

2 fruits and 2 cheeses
2 * 1 * 6 * 5 = 60

Total = 72
Do I need to divide here by number of things chosen b/c order doesn't matter? Pls advise.

[kvcpk]

6 cheeses and 2 fruits

If 1 fruit and 1 cheeseare on platter:
fruit can be chosen in 2c1 = 2 ways
cheese can be chosen in 6c1 = 6 ways.
Total 2* 6 = 12 [since both should occur]

If 2 fruits and 2 cheese are on platter:
fruit can be chosen in 2c2 = 1 ways
cheese can be chosen in 6c2 = 15 ways
total = 15*1 = 15 ways

Any of these should occur. Hence 15+12 = 27 ways..

Hope this helps!!

[san2009]

kvcpk, not sure if u read my post
but i am trying to solve this Q with the SLOT method
thx

[kvcpk]

kvcpk, not sure if u read my post
but i am trying to solve this Q with the SLOT method
thx

I am sorry.. I read that But I am unaware of SLOT method. Wil leave it for others to take it thru.

[GMATGuruNY]

A certain restaurant offers 6 kinds of cheese and 2 kinds of fruit for its desert platter. If each dessert platter contains an equal number of kinds of cheese and kinds of fruit, how many different dessert platters could the restaurant offer?

a. 8
b. 12
c. 15
d. 21
e. 27 is OA

My take - pls tell me how to correct my "SLOT" method approach
1 fruit 1 cheese
2 * 6 = 12

2 fruits and 2 cheeses
2 * 1 * 6 * 5 = 60

Total = 72
Do I need to divide here by number of things chosen b/c order doesn't matter? Pls advise.

Yes. When order doesn't matter, we need to divide by the (number of slots)!:

Two make a plate with 2 cheeses and 2 fruits:
A combination of 2 fruits from 2 choices: (2 * 1)/(2 * 1) = 1 (2 slots, so we divide by 2! = 2*1)
A combination of 2 cheeses from 6 choices = (6*5)/(2*1) = 15 (2 slots, so we divide by 2! = 2*1)
Now we need to combine our fruit choices with our cheese choices: 1*15 = 15.

To make a plate with 1 cheese and 1 fruit: 2*6=12.

Total possible combinations = 15 + 12 = 27.

The correct answer is E.

[san2009]

Thanks Mitch. Very helpful!
I had a follow-up question though.

When order does NOT matter - we are supposed to divide by (number of slots)!
Although I understand the reason why...I'm trying to understand that a bit better still
You divided each sub category by the (number of slots)!
Why is that?
Is that always how it is?

I read somewhere that we're supposed to divide by number of things chosen
which is why I thought maybe we are supposed to divide the first scenario (1 fruit 1 cheese) by 2!
and the second scenario (2 fruits 2 cheeses) by 4!.

Thanks again!!

[GMATGuruNY]

Thanks Mitch. Very helpful!
I had a follow-up question though.

When order does NOT matter - we are supposed to divide by (number of slots)!
Although I understand the reason why...I'm trying to understand that a bit better still
You divided each sub category by the (number of slots)!
Why is that?
Is that always how it is?

I read somewhere that we're supposed to divide by number of things chosen
which is why I thought maybe we are supposed to divide the first scenario (1 fruit 1 cheese) by 2!
and the second scenario (2 fruits 2 cheeses) by 4!.

Thanks again!!

In this problem, we're combining from different sources (our fruit source and our cheese source).

We need to deal with each choice separately.
Since order doesn't matter, to determine the number of combinations possible from each source, we need to divide by (the number of elements being chosen)!:

From our cheese source, we have 6 choices.
A combination of 2 cheeses requires 2 slots, so we divide by 2!: (6*5)/(2*1) = 15.
A combination of 3 cheeses would require 3 slots, so we would divide by 3!: (6*5*4)/(3*2*1) = 20.
A combination of 4 cheeses would require 4 slots, so we would divide by 4!: (6*5*4*3)/(4*3*2*1) = 15.

From our fruit source, we have 2 choices:
A combination of 1 fruit requires 1 slot, so we divide by 1!: 2/1 = 2.
A combination of 2 fruits requires 2 slots, so we divide by 2!: (2*1)/(2*1) = 1.

So to make a plate with 3 cheeses and 2 fruits, we would multiply the results above: 20*1 = 20 possible combinations.

Does this help?

[san2009]

yes, that is helpful. i understand. Each sub-category, whether that is cheeses or fruits need to be dealt with separately, when considering the order. Thanks!