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by knight247 » Fri Sep 30, 2011 3:33 am
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
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by shankar.ashwin » Fri Sep 30, 2011 4:55 am
There should be a minimum of 8 tickets to give each of the cousins 2 tickets.

These 8 tickets can be distributed only in 1 way ( 2 each).

Now neglect this 8 from total and see which answer choice satisfies 120.

I started with C (15-8 = 7)

7 tickets to be distributed to 4 people -- 120 ways.

C IMO

Not very sure though
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by GMATGuruNY » Fri Sep 30, 2011 7:48 am
knight247 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/experts-any- ... 82307.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.
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by phoenix111 » Fri Sep 30, 2011 10:23 pm
knight247 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
Answer should be 15 ( Option C )

Total Voucher : X

Give 1V to each nephew : remaining X - 4

Now each nephew must get alteast one more voucher,
so the problem reduces to

w + x + y + z = X - 4 ( w,x,y,z >= 1 )

Number of solutions : (X-4 - 1)C3 = 120
Instead of solving we can subsitute the options.

B: 15 will give (15 - 4 -1)C3 = 10C3 = 120.
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