In how many ways can 5 different candies be distributed in

Thanks Eva. A quick question - is there any shortcut method for this question? Appreciate your help. If GMATClub had allowed, I would have given you 100 kudos. Thanks for helping me.

Thanks

Here, I don't see a shortcut, each case must be treated separately.

Hi,
Sorry, i have a hard time with this topic,
Do you mean that for
3-1-1-N we don t need 2C1?
2-2-1-N we don t need 3C1?
I stll don t understand well the reason...

Could you please explain it in other words?

Thank you

R26

Posted from GMAT ToolKit

Each one is explained in words (see text in blue above).

Thank you very much, it is clear now ^_^
(By the way, is it a typo in red for the 2-1-1-1? You wanted to say 3! But though directly of the result being 6.)

Thank you again all of you

R26

Oh yes, I edited to correct the 6! to 3! = 6.

Is this the only way to solve this problem? tricky question I misunderstood it completely..

We just determine the distribution of specific candies to a grouping of 5,4,2,3, or 1 per basket but there is no need to arrange the distribution on to the baskets since the containers are identical.

5-0-0-0: \(\frac{5!}{5!0!} = 1\)
4-1-0-0: \(\frac{5!}{4!1!}*1 = 5\)
3-2-0-0: \(\frac{5!}{3!2!}*\frac{2!}{2!} = 10\)
3-1-1-0: \(\frac{5!}{3!2!}*\frac{2!}{1!} * \frac{1!}{1!} = 20\) Then, we divide by 2! since 1 and 1 even interchanged doesn't matter. \(\frac{20}{2}=10\)
2-1-1-1: \(\frac{5!}{2!3!}*\frac{3!}{1!2!}*\frac{2!}{1!1!}*1=60\) Then, we divide by 3! since 1,1 and 1 even interchanged doesn't matter since the baskets are identical. \(\frac{60}{3!}=10\)
2-2-1-0: \(\frac{5!}{2!3!}*\frac{3!}{2!1!}=30\) Then divide by 2! since 2 and 2 even interchanged doesn't matter. \(\frac{30}{2}=15\)

Answer: \(1+5+10+10+10+15=51\)

I dont quite understand why we are multiplying combinatorics (ie 5C3*2C2, etc.)?
Is there some general rule or application on when I should do this?

Also in the last sequence 2-1-1-1, you do 5C2 and then 3! which I'm assuming is just 3C1. Why isn't it 5C2*3C1*2C1*1C1? (Appreciate that answering the above may explain this).

Thanks in advance

When two selection are made for the same case, we multiply (the AND rule)
Say, in how many ways can you select a meal which includes a burger and a beverage. There are 2 different kinds of burgers and 3 different kinds of beverages. You can do it in 2C1 * 3C1 ways. You have to select a burger AND a beverage so you multiply their respective selections and get the answer as 6.

On the other hand, in how many ways can you select a meal of either a burger or a pizza but not both? There are 2 different kinds of burgers and 3 different kinds of pizzas. You can do this in 2C1 + 3C1 = 5 ways. You add here because you have to select burger OR pizza. When you have OR, you add.

Here you have to select 3 candies for one basket AND 2 candies for another basket so you do 5C3 * 2C2

Yes, 3*2*1 at the end is 3C1*2C1*1C1.

i am not sure why the below approach is not working for this one ..
let say there are 4 Kids , and we want to distribute 5candies among them .
CCCCC||| is one possible arrangement , here the first kid got all the 5 candies.
so why cant we use \(^8C_3\) to get 56 as answer.

Why not E like here? https://gmatclub.com/forum/how-many-way ... 37198.html
Each marble has 4 options so 4^5?

Bunuel VeritasPrepKarishma can you help me please?
I am getting confused by these distribution problems. Can you help me differentiate the formulas I must use in each situation?

I had:

- for x different objects distributed among n people: n^x (which i thought was the case here in this question, and still didn't understand why not)

-for x different rings among n different fingers: use bars, number of bars = n-1, total ways = (x+bars)!/bars!
(here, i didn't understand how fingers were treated differently from people, since in both cases they are different among themselves, that is, different fingers but also different people)

-for x identical objects among n different things: here as x also repeats, use formula as above but dividing by x!. that is, total ways= (x+bars)!/x! bars!

- for x identical objects among n different things, so that each n receives at least one object:
x-1Cbars
Here, for example, 20 fruits among 4 people, so that each receives at least one = 19C3= 19!/16!3!
(here i can't see what the 16! would represent. in the case where people could get 0 fruits, you would divide by the repeated bars (3!) and the repeated fruits (20!), but i can't get what the 16! would be. is it because first you distribute 4 and then all other 16 would be repeated?

Thank you very much in advance. And thank you for all your genius posts, I am learning a lot from you both

why cant it be 4^5 ? whats different https://gmatclub.com/forum/in-how-many- ... 31187.html ? tx

Since the Candies are Different ---> but the Baskets they are Distributed to are IDENTICAL, it only matters which candies are chosen to end up in a particular Number Stack. It does not matter which Basket gets that Number Stack.

There are 6 Different Scenarios in which the 5 Different Candies can be Divided among the 4 Identical Baskets:

Scenario 1: 0 - 0 - 0 - 5

"5 Choose 5" = 1 Way in which we can choose the 5 Different Candies. It does NOT matter which Basket receives them.

Scenario 2: 0 - 0 - 1 - 4
"5 Choose 4" and "1 Choose 1" = 5! / 4! * 1! = 5 Ways in which we can choose the 4 Different Candies for 1 Stack and 1 Candy for the Other Stack.

Scenario 3: 0 - 0 - 2 - 3
"5 Choose 3" and "2 Choose 2" = 5! / 3! * 2! = 10 Ways in which we can choose the 3 Different Candies for 1 Stack and 2 Candies for the Other Stack

Scenario 4: 0 - 1 - 1 - 3
"5 choose 3" and "2 Choose 1" and "1 choose 1" =
5! / 3! * 1! * 1! = 10 Ways

Scenario 5: 0 - 1 - 2 - 2
"5 Choose 2" and "3 Choose 2" and "1 Choose 1" =

However, since 2 of the Identical Stacks are of EQUAL SIZE (2 and 2), we will have Over-counting. For Each 1 Distribution we want counted, 2! will actually be Counted. Therefore, we must Divide by 2! to Remove these Repeats.

5! / 2! * 2! * 1! * (1 / 2!) = 5! / 2 * 2 * 2 = 5 * 4 * 3 / 2 * 2 = 15 Ways

Scenario 6: 1 - 1 - 1 - 2
"5 Choose 2" = 5! / 2! * 3! = 10 Ways



Adding up the 6 Scenarios =

1 + 5 + 10 + 10 + 15 + 10 = 51 Different Ways in which we can Distribute 5 Different Candies to 4 Identical Baskets

Hi EvaJager, I saw that you had given an explanation for this similar distribution question as well: https://gmatclub.com/forum/how-many-way ... 37198.html

What is the difference between the wording of this question and the other linked question? I don't see anything significant and want to improve my understanding of such questions. Could you please elaborate why this solution is much more involved?

Hi Bunuel and KarishmaB, I'd like to +1 the post from Juliaz. It is indeed confusing when to apply the methods used for these kinds of distribution problems. I have provided an identical example above that has a much simpler solution path, so I'm also trying to better understand when to use what approach to solve these problems efficiently.

To this end, I think a breakdown of when to use the calculation tactics mentioned in in Julia's post can help us all understand this a bit better. I am also curious if Julia's list is exhaustive or if there are other calculation techniques that we should be aware of. Please let us know if you can help!

Thanks, Kaushik