Problem Solving

Mrs. Smith 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. Smith 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

OA is C

We need to distribute atleast "2" to each nephew.. So lets say we have "N" ..
Now, distribute "8" of "N" ..
So, now each nephew has two tickets..

So, we are left with N-8 .. let N-8 as "A"

Using Seperator method
We need to distribute "A" among 4

So..

(A+3)!/3!*A! == > (A+3)(A+2)(A+1) = 720
=> (A+3)(A+2)(A+1) = 8 x 9 x 10

So, A = 7
N = 7+8 = 17

Answer [spoiler]{C}[/spoiler]

suppose Mrs Smith have N tickets

now each must have at least 2, N-8 remaining tickets can be bistributed such that any one could have 0,1-- or all remaining tickets

seprator method :- suppose N-8 = K
K+3C3 = 120
(K+3)(k+2)(k+1)= 6*120 = 720
720 must be multiple of 5
lets try for 5*6*7=210, next possible smaller triplet with multiple of 5, 8*9*10, Bingo
K+1 = 8, K = 7, N = 8+7 = 15
C

rakeshd347 wrote:Mrs. Smith 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. Smith 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

To guarantee that each nephew receives at least 2 vouchers, first give each of the 4 nephews EXACTLY 2 vouchers.
This accounts for 8 of the vouchers.
Now that each nephew has 2 vouchers, there are no restrictions on how the REMAINING vouchers can be distributed.
Thus, 120 = the number of ways to distribute the REMAINING vouchers.

We can plug in the answers and use the SEPARATOR METHOD to count the number of possible distributions.
For an explanation of the SEPARATOR METHOD, please check the following links:

https://www.beatthegmat.com/combinations-t120668.html
https://www.beatthegmat.com/inserting-st ... 67423.html
https://www.beatthegmat.com/how-many-non ... 34635.html
https://www.beatthegmat.com/lets-have-a- ... 69973.html

Answer choice C: 15 vouchers.
Remaining vouchers after the first 8 have been distributed = 15-8 = 7.
Let the 7 vouchers = VVVVVVV.
Since these 7 vouchers can be distributed among up to 4 nephews, we need 3 separators: |||.
Number of ways to arrange the 10 elements VVVVVVV||| = 10!/(7!3!) = 120.
Success!

The correct answer is C.

GMATGuruNY wrote:

rakeshd347 wrote:Mrs. Smith 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. Smith 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

To guarantee that each nephew receives at least 2 vouchers, first give each of the 4 nephews EXACTLY 2 vouchers.
This accounts for 8 of the vouchers.
Now that each nephew has 2 vouchers, there are no restrictions on how the REMAINING vouchers can be distributed.
Thus, 120 = the number of ways to distribute the REMAINING vouchers.

We can plug in the answers and use the SEPARATOR METHOD to count the number of possible distributions.
(For an explanation of the SEPARATOR METHOD, please check here: https://www.beatthegmat.com/combinations-t120668.html.

Answer choice C: 15 vouchers.
Remaining vouchers after the first 8 have been distributed = 15-8 = 7.
Let the 7 vouchers = VVVVVVV.
Since these 7 vouchers can be distributed among up to 4 nephews, we need 3 separators: |||.
Number of ways to arrange the 10 elements VVVVVVV||| = 10!/(7!3!) = 120.
Success!

The correct answer is C.

Hi Mitch,

The link you have attached can't be found. Thats the error I am getting. Can you please send me some link for separator method. I have no idea what so ever what is this separator method.

I've fixed the link (and added a few more). Please revisit my post above.

Rakesh,
This problem is super cool what is the source. I like to get the same.

Regards,
Uva.