Calculating Gem Distribution in a Combination and Permutation Problem

[haoku]

Homework Statement

There are 10 different coloured gems will be given to 6 students according to their marks.
The marks of the students are 60%,12%,12%,12%,12%,12%.
The number of gems obtained is according to their marks.
How many ways can it distribute the gems?


2. The attempt at a solution
According the the ratio the number of gems obtained for each student is 1,1,1,1,1 and 5.
But I don;t know how to calculate number of ways to distribute the gems.

 

[Sleek]

Is 210 the answer?

I'm assuming that each student receives atleast one gem. So the 5 students with 12% marks can get only 1 gem each. They cannot get more than 1, i.e. 2, because then each of the 5 people must be given 2, and nothing would be left for the 60% student (who should get more).

Since 10 gems (of different color) have to be given among 6 students, the combination would be 10C6 = 210. The gems can be arranged only in these many ways, since the 60% guy cannot get less than 5 gems and the others can get more than 1, assuming each gets atleast one. If it is possible that the 12% guys don't get any gems, only one more possible combination is added, so the answer is 211.

Regards,
Sleek.

 

[haoku]

I have asked the teacher and the teacher said each student who got 12% mark will get 1 gems and student with 60% will have 5 gems.
Should the answer be 10C5?

 

[HallsofIvy]

Put the students in a line in that order: the 5 who got 12% first then the person who got 60% (sounds like one of my classes!).
How many ways can you choose a gem to give the first student? After you have done that (and have 9 gems left), how many ways can you choose a gem to give the second student? The third? Fourth? Fifth? By this time you have 5 gems left to give the last student. No choices, just hand them to the fifth student.

The hard way: let the student who got 60% get his/her gems first: How many ways can you choose 5 out of 10 gems to give him/her? Now you have 5 gems left, how many choices of a gem to give the next? And so on. It's interesting that that gives exactly the same answer.

 

[haoku]

You mean easy one : 10*9*8*7*6
=30240
hard one=10C5*5*4*3*2*1=30240 also!
If order is not important, should I divide the answer by 5!?

 

[Sleek]

Ah! Great solution HallsofIvy!

You may have to consult your teacher on that I guess. If the question is about distributing the gems to each of them one by one, then 30240 is right.

If the question is in how many ways can the gems be arranged among the five students, then you have to divide by 5!. But in that case, you may not take this route towards the answer. The solution I posted previously neglected a very important fact that only ONE person gets 5 gems and the rest 1. So in that case,

We have to pick out 5 gems for the 60% guy out of 10, 10C5=252. The remaining 5 has to be distributed among the 5 students, i.e. 5C5=1.

Thus the total ways is 252, i.e the same as 30240/5! = 252.

 

[haoku]

Thanks