Selecting Identical items

[Joepc]

There 4 Apples in a Basket ,5 Mangoes in another basket ,2 Kiwis in another basket in a fruit shop

consider each fruits in the basket are of same colour, shape , size and weight another way to say fruits of its kind in the baskets are identical

1)In how many ways I can select at least one fruit

5*6*3 = (90-1) 89 ways[ where 1 is non selection ]

2)In how many ways I can select so that one fruit is selected from each fruit basket

4*5*6 = 90 ways

Above all are bookish way to get answer, I failing to understand how did they came into these numbers

As per the rule for selecting identical items :-

There only one way to select identical for any number of selection

As per the above statement when there are 4 apples of same kind in a basket

The ways of selecting apples are

Case 1:-
Selecting 4 apples
Selecting 4 apples is one way
and not selecting 4 apples is another way, hence it is 2 ways.

Case 2 :-
Similarly selecting 3 apples
Selecting 3 apples is one way
and not selecting 3 apples is another way, hence it is 2 ways.

Similarly selecting 2 apples
Selecting 2 apples is one way
and not selecting 2 apples is another way, hence it is 2 ways.


Similarly selecting 1 apples
Selecting 1 apples is one way
and not 1 apple selecting is one way, hence it is 2 ways.

Hence the total ways is 2 to the Power 4 and it is 16 ways
which is same as of selecting 4 different numbered\Weighted Apple or distinct type of apples in the basket

Getting confused here, can somebody help?

[ceilidh.erickson]

I'm pretty confused by what you've written here, too. Are you quoting from a source, or asking whether this understanding is the correct one? Either way, I'll explain, based on what I think you're saying here - one question at a time.

1)In how many ways I can select at least one fruit

5*6*3 = (90-1) 89 ways[ where 1 is non selection ]

The idea here is that we want to count every scenario from [1 apple, no mangoes or kiwi] up to [4 apples, 5 mangoes, 2 kiwi]. We can't choose more than that.

So, we want to multiply the possibilities for each type of fruit. You might think that would be 4*5*2, but consider that NO APPLES is also a possibility for what you could pick to satisfy the question. So, you can have:
0, 1, 2, 3, or 4 apples: 5 possibilities
0, 1, 2, 3, 4, or 5 mangoes: 6 possibilities
0, 1, or 2 kiwis: 3 possibilities
5*6*3 = 90

The trick is to remember that we have to pick AT LEAST ONE fruit. One of those 90 scenarios was the 0, 0, 0 scenario. So we have to subtract that out: 90 - 1 = 89.

[ceilidh.erickson]

2)In how many ways I can select so that one fruit is selected from each fruit basket

4*5*6 = 90 ways

Here's where I've lost you. If we're treating all of the fruits in a given basket as identical, then there should only be ONE way to pick one fruit from each basket: pick an apple, pick a mango, pick a kiwi. 1*1*1 = 1

Only if we're treating the apples as different do we bother with thinking about Apple 1, Apple 2, etc. If the question asked about UNIQUE combinations of individual fruits, we'd then multiple 4*5*2.

I have no idea mathematically where the "6" came from in 4*5*6.

I also have no idea how on earth we can arrive at the answer 4*5*6 = 90. It equals 120. There's no way of dividing 120 by anything to get 90.

So if you did get this out of some sort of book or manual, I'm as baffled as you are. What is the source? (Please also note that if you are indeed quoting directly from some source, this source is not written in grammatically correct English, and you should probably not use it).

[ceilidh.erickson]

As per the above statement when there are 4 apples of same kind in a basket

The ways of selecting apples are

Case 1:-
Selecting 4 apples
Selecting 4 apples is one way
and not selecting 4 apples is another way, hence it is 2 ways.

Case 2 :-
Similarly selecting 3 apples
Selecting 3 apples is one way
and not selecting 3 apples is another way, hence it is 2 ways.

Similarly selecting 2 apples
Selecting 2 apples is one way
and not selecting 2 apples is another way, hence it is 2 ways.


Similarly selecting 1 apples
Selecting 1 apples is one way
and not 1 apple selecting is one way, hence it is 2 ways.

Hence the total ways is 2 to the Power 4 and it is 16 ways
which is same as of selecting 4 different numbered\Weighted Apple or distinct type of apples in the basket

This counting method doesn't make any sense. If you want to know how many ways there are to select 4 apples out of 4, there is 1 way: select all 4 apples. We don't count NOT selecting 4 apples as a way to select 4 apples. That's nonsensical.

"Not selecting 4 apples" = selecting 0 apples. But semantically, it seems to me that "not selecting 3 apples," "not selecting 2 apples," etc, are also equal to 0. So you've counted the 0 case 4 times.

Then for some inexplicable reason, those counts were multiplied. If you are counting scenarios and identify DISTINCT cases that satisfy the prompt, you ADD the scenarios.

For example, if we wanted to select at least one person to serve on a committee, out of 5 eligible people, we could say:
case 1 - 1 person selected: there at 5 ways to choose 1 out of 5.
case 2 - 2 people selected: there are (5!)/(2!)(3!) = 10 ways to choose 2 out of 5, where order doesn't matter.
case 3 - 3 people selected: there are (5!)/(2!)(3!) = 10 ways to choose 3 out of 5, where order doesn't matter.
case 4 - 4 people selected: there are (5!)/(4!)(1!) = 5 ways to choose 4 out of 5, where order doesn't matter.
case 5 - all 5 people selected: there is only one way for this to happen.
Total = 5 + 10 + 10 + 5 + 1 = 31 possible ways.

We only count this way when we're treating the members of the group as individuals. This will always be true when you're dealing with people. When you're dealing with inanimate objects (apples, $5 bills, marbles, etc), we don't think of picking different individual objects as separate scenarios.

[Joepc]

Thanks Erickson for your Reply.

It was a type typo error for the second question it is 4*5*2 = 40 ways

For the question

There 4 Apples in a Basket ,5 Mangoes in another basket ,2 Kiwis in another basket in a fruit shop
and the fruits in the each kind baskets are identical

2)In how many ways I can select so that atleast one fruit is selected from each fruit basket

And below is my reasoning

For Apples

One (here one is Quantity) Apple can be selected from the identical 4 identical apples = 1 Way
Two(here two is Quantity) Apple can be selected from the identical 4 identical apples = 1 Way
three(here three is Quantity) Apple can be selected from the identical 4 identical apples = 1 Way
four(here four is Quantity) Apple can be selected from the identical 4 identical apple = 1 Way
Hence the total number of ways of selectig 4 apples is 4 ways

In a Similar Way
The total number of ways of selecting 5 Identical Mangoes is 5 ways
The total number of ways of selecting 2 identical kiwis is 2 ways

Hence the total number of ways of selecting 4 identical Apples, 5 identical Mangoes and 2 identical kiwis are
4*5*2 = 40 ways.


Your Question on How did i Got this question, It was from a reasoning of below question

A florist Had

5 roses of distinct color
6 white lilies(saying that lilies are identical)
What is the Number of ways one can purchase such that at least one rose and lily are bought

Case 1) Roses :-

Since roses are Distinct colored the total number of Ways are

Formula way :- 5c1+5c2+5c3+5c4+5c5
Thought\Reasoning way :- One rose can either be purchased or not purchased there are two possibilities for each distinct colored roses hence its 2 to the power 5 which is the total number of ways of buying 5 distinct colored roses

When it comes to atleast one roses to be bought the way will be (2 to the power 5 - 1) = 31 ways

Case 2) lilies :-

Since lilies are identical
As per the above reasoning i had for apples, the total number of ways to way at-least one Lilly from 6 identical lilies are in 6 Ways

hence the answer is

= 31 *6
= 186 ways

And if the question is atleast one flower to be selected

For all 5 different\Distinct Roses it is same 2 to the power 5 ways
For all 6 identical lilies it is 7 ways(6 ways of selecting and one way of not selecting)

= (2 to the power 5 * 7) -1
= 223 ways

So my doubt in the case of 4 identical apples, you can select or not select each appple in the 4 identical apples in 2 ways it should be 2 to the power 4 but it is 5 ways (4 ways for selecting and 1 way for not selecting) hence the total number of ways of selecting 4 identical apples are are 5 ways why not 2 to the power 4?

[ceilidh.erickson]

First of all, I've never seen a real GMAT question that treated some objects in a category (e.g. flowers) as distinct, and others as identical. It's not mathematically wrong to do so, but it's confusing. I doubt you'll ever see it.

Once you've corrected the typo, this is a much more logical question:

2) In how many ways I can select so that at least one fruit is selected from each fruit basket

This is now just a simple variation on your question #1. The difference, though, is that 0 is no longer an option for the number of any given type of fruit. So instead of:
5*6*3
we now have:
4*5*2

In other words, you can select:
1, 2, 3, or 4 apples: 4 possibilities
1, 2, 3, 4, or 5 mangoes: 5 possibilities
1, or 2 kiwis: 2 possibilities
4*5*2 = 40

This is the total number of way to select at least one item from each basket. It's not, as you said, "the total number of ways of selecting 4 identical Apples, 5 identical Mangoes and 2 identical kiwis." If we were selecting ALL of those, there would only be 1 single way to do that: select all of them.

To your question:

So my doubt in the case of 4 identical apples, you can select or not select each appple in the 4 identical apples in 2 ways it should be 2 to the power 4 but it is 5 ways (4 ways for selecting and 1 way for not selecting) hence the total number of ways of selecting 4 identical apples are are 5 ways why not 2 to the power 4?

When the apples are identical, there is no such thing as "each" apple. You can pick no apples, 1 apple, 2 apples, 3 apples, or 4 apples. We don't count for EACH apple whether it's chosen or not. The math you're citing (2^4 - 1) is only for picking "at least one" out of a DISTINCT group of items.

Consider this analogy:

If I have five $1 bills, and I give some amount of that money to you, how many different possibilities are there for the amount of money that I could give you?

Well, I could give you $1, $2, $3, $4, or $5. You wouldn't run a separate analysis on the possibility of each individual bill being chosen, because it wouldn't matter.

Does that help?

[Joepc]

Thanks Erickson,

In the case of Flowers question , I may be overthinking of the possible distinct and identical scenarios

With this interaction i got some great insight in identical and distinct items, And thank you very much for your interest in replying.

[ceilidh.erickson]

My pleasure!

To make a very broad observation... I have found that the level of difficulty and complexity of combinatoric & probability questions that exists on the internet (on this forum, on test prep companies' sites, etc) far exceeds what I've found on the actual test. These topics are complex enough that they're already high-difficulty, so you won't find too many problems that add excruciating twists and turns on top of that (or at least not until you're reaching the 750+ level). But I've seen a lot of practice problems that I jokingly refer to as "900-level" problems in these topics!

My advice: get the basics down, but don't worry about the complexities too much. Focus on mastering the question types that seem easier in theory, but that you'll see more of - percents, divisibility, etc.

[Matt@VeritasPrep]

To piggyback on Ceilidh's last response, the GMAT is NOT in the habit of asking very technical questions that are actually simple if you know an advanced formula or two. Instead, the GMAT likes to ask questions that "anyone" could answer with a little ingenuity and common sense, but that ARE NOT formulaic in the least, and aren't much easier for someone with a background in mathematics than for someone with a background in painting. This question is routine for someone who's taken an intro class on Prob and Comb, but opaque for anyone who hasn't.

I often have students in my classes who haven't done math since high school who turn out to be much better than at GMAT math than engineers and computer scientists, and this is due to the nature of the questions the GMAT asks: they're designed to probe your deeper reasoning ability, not your prior mathematical training.