Mrs. Napueta has been given film vouchers. Each voucher

Question

Mrs. Napueta has been given film vouchers. Each voucher allows the holder to see a film without charge. She decides to distribute them among her four nephews so that each nephew gets at least two vouchers. How many vouchers has Mrs. Napueta been given if there are 120 ways that she could distribute the vouchers?

(A) 13 (B) 14 (C) 15 (D) 16 (E) more than 16

mailtheguru · Answer

13.

Give 2 vouchers to each of the nephews. (8)

The rest can be distributed in n * n-1 * n-2 ... ways = 120, hence n=5

5+8=13

paddyboy · Answer

Yes, but only under the assumption that each of the nephews does get an extra ticket. What if she gives her favourite nephew all of the remaining 5 tickets!  Good point! 

All said and done, the only plausible answer is 13. If you consider the case I just described above, the number of combos is just too many...

Mikhail Bondarenko · Answer

I'll go for D=16. Here's why:

Let n be the total number of vouchers. Let's minimize the number of vouchers per child and assume each child gets m= 2. How large then should n be for the total number of combinations C to equal 120:

C = n!/(n-m)!m!
120 = (n-2)(n-1)*n/(n-2)!2! // factoring the factorial
120*2!=(n-1)*n // canceling (n-2)!, multiplying both sides by 2!
240= n^2-n // 2!=2
n^-n-240=0 
D = -1^2 - 4*1*(-240) = 961 = 31^2
n = (1+/- sqrt (31^2))/2
n = - 15 or n = 16. Obviously, n is positive (number of vouchers). 
//checking 16^2 - 16 - 240 = = 0

n is consistent with each child getting at least 2 vouchers and the total combinations being 120

(if we assume that m >2 then the number of combinations will exceed 120 since since the combination function grows in m (plug in the numbers)... n=16 looks the unique solution to the above, anything more or less will not give the exact amount )

Any ideas?

cicerone · Answer

Answer must be 13.
Folks, u need a little bit of P & C here.
Soon I am going to start a thread on P&C and all the concepts regarding questions liike these will be discussed there...........

Untill then byeeeeeeeee

kevincan · Answer

How do you conclude that the answer must be 13?

Mikhail Bondarenko · Answer

how did I get the formula in bold? Mystatistics textbooks gives the following: https://en.wikipedia.org/wiki/Combination

kevincan · Answer

That's what I got! How did you get it?

I thought as follows- give each nephew a voucher and put the other n-4 vouchers in a row, with spaces between adjacent vouchers, a total of n-5 spaces. Choose three of the spaces: nephew A gets the vouchers to the left of the leftmost space chosen, nephew B gets the vouchers between the two leftmost spaces, and so on. 

so the number of ways of assigning these indistinguishable vouchers is a combination of size 3 of n-5 spaces.

So (n-5)(n-6)(n-7)/3!=120 Thus n-5=10 and n=15

cicerone · Answer

That's it kevin, I solved using the same concept and i think this is the only correct and practical way of answering this......

By the way, for the last 6 months i am not active because of some health problem...I am back and u can expect active participation from me for atleast the queries that you post.....

Keep rocking

jayt696969 · Answer

formula for combinations: nCk = n! / ((n-k)! * k!)

so 4C4 = 4! / ((4-4)! * 4!) = 1

But nonetheless, I don't even think that is the right formula for this specific example. Think about this: If you have 1 voucher, and 4 nephews, you can give it to any one of them, so there are 4 ways. If you have 2 vouchers, you can give the first to any of them and the second to any of them, for a total of 16 ways. 
Given: n = # vouchers 
Number of ways to distribute n vouchers to four nephews: 4^n

In each of the 120 ways, each nephew has at least two. So if you take two from each, each of the 120 ways are still unique. Thus you can say that the amount of vouchers is n-8 and that the formula is 4^(n-8) = 120.

However this is where i am stuck, none of the answers work for my formula. If someone can please point out where i erred, thank you very much.

vshaunak@gmail.com · Answer

simpler way to solve such problems is by using the formula - 
The no. of ways to distribute n items among r people where each gets zero or more items is (n+r-1) C(r-1).

In this question - let's say the number of vouchers = n
n-8 vouchers to be distributed among 4 people. where each get zero or more.
So total ways = (n-8+4-1)C3 = 120
solving this (n-5)(n-6)(n-7) = 120 * 3! = 720 
 hence n = 15

Bluebird · Answer

I got E as my answer and this is how:

# of vouchers = n-8 (total # of vouchers - 2 for each of 4 nephews)

So, we have to figure out the total number of combinations for the remaining n-8 vouchers among the 4 nephews.

(n-8)C4--> (n-8)!/(4!*(n-8-4)!) = (n-8)!/24*(n-12)!

Set this equal to 120 and multiply by the 24 and you get

(n-8)!/(n-12)!=2880

Now just plug in numbers to see which satisfies the equation.

If you plug in 16 you get an imbalanced equation: 1680 vs. 2880

So the answer must be E, more than 16 vouchers.

What's the OA?

vijay2001 · Answer

What is the source of this question? 

I)In the above, each nephew is eligible for all the vouchers 
i.e. n*n*n*n = n^4 =120 ??which is not possible

II) Another way, is considering this as a permutation, 
nP4 =120 => 5
Total is 8+5 =13

I is the correct way of solving.

Mrs. Napueta has been given film vouchers. Each voucher

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