Permutation and combination

Question

Your mother-in-law buys 1000 small gifts to give to relatives for Christmas. Each of the 1000 things in different. There are 25 relatives to give gifts to. How many ways are there to distribute the gifts?

 

The correct answer is $25^{1000}$.

I always get confused in these type of question like why it can't be $1000^{25}$?

I know some people may feel its silly question but please trust me many people like me are confused with this doubt.

Answer 1

Gift 1

Gift 2

Gift 3

Gift 4

Gift 5

...

...

Gift 1000

 

• Every gift has 25 possibilities; and there are total 1000 gifts.
• thats why 25*25*25*.... 1000 times
• so answer is 25^1000. 

Personally this question is ambiguous for me.  

 

Comments

Can't we think like this?
1st person has 1000 gift options.

Then $2^{nd}$ person also has 1000 gift options.

then $3^{rd}$ person also has 1000 gift options

like that we have 25 people so we have the total no. of ways = $1000^{25}$

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No thats a wrong way of thinking it.If u r considering that the first person has 1000 choices then the second person will definitely have less than or equal to 1000.Moreover you have to choose a relative out of the 25 relatives to make it happen which will be ²⁵C₁.

Lets consider we have only 3 relatives R1,R2 and R3and 5 toys T1 to T5.Now at the first place u will choose a relative in 3C1 ways then he can have any of the 5 toys.But dont forget that person may or may not choose a toy as well.
Now lets say we have a distribution as follows :

	R1	T1
	R2	T2,T3
	R3	T4,T5

Here 3C1 ways to choose R1 then 5C1 ways for him to choose a toy.Then we choose R2 by 2C1 ways and give him to choose toys of which he has 4C2 posibilities of choosing T2,T3.Lastly for R3 we neither need to choose him nor the toys for him.
This makes it 3*5*2*6*1*1
But this is an overcounting we can choose R2 first also as 3C1 and give him the two toys by 4C2 and so on fo the rest which is being counted 3! times in the above expression obtained .How 3!? Just choosing the different orders of the three relatives.

For in depth understanding of distinct to distinct distributions u can follow Deepak Sirs combinatorics free on GoClasses Website.
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Answer 2

Hii, for such type of questions always remember that while distributing gifts to relatives there might be a relative which will recieve no gifts at all, but there would be no gift which doesnt get distributed.

Basic principle of distributing is that you distribute all you have, for here we are distributing 1000 gifts so we will distribute them full.

So each gift can go to any of the 25 relatives i.e for each gift there are 25 possibilities.

Therefore No of ways = 25 X 25 X 25 ........... 1000 times.

Hence total ways = 25^1000.