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Mrs. Smith has been given film vouchers. Each voucher allows

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Re: Mrs. Smith has been given film vouchers. Each voucher allows  [#permalink]

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New post 26 Feb 2019, 06:43
Debashis Roy wrote:
Bunuel VeritasKarishma
Why dont we consider the no of ways in which 8 vouchers are to be distributed among the 4 nephews equally? Isnt that also included within 120 ways?

Please explain


Note that the vouchers are not distinct. Since each nephew gets at least 2 vouchers, we give away 2 to each in 1 way. The 8 vouchers are distributed among 4 nephews in 1 way only (2 vouchers per nephew).
The extra 7 vouchers (since we have a total of 15) need to be distributed among the 4 nephews such that any nephew may get any number of vouchers. This can be done in
10!/7!*3! = 10*3*4 = 120 ways

Answer (C)
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Re: Mrs. Smith has been given film vouchers. Each voucher allows  [#permalink]

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New post 27 Feb 2019, 10:11
Crack700 pls how did n-5C3=120 give 15? How did you solve for n from that formular to give 15??? Can anyone help? I'm lost please.

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Re: Mrs. Smith has been given film vouchers. Each voucher allows  [#permalink]

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New post 27 Feb 2019, 10:15
[quote="crack700"][quote="bogos"]It took me 10 minutes to solve it :(
First, it is remarked that all vouchers are equivalent.
Let's call nephews A, B, C, D and their number of vouchers (2+a), (2+b), (2+c), and (2+d), respectively, where a, b, c, d non negative integers (since each of them has at least 2 vouchers).
If n is the total number of vouchers, then:
a+b+c+d=n-8 or a+b+c+d=m with m=n-8
Let X=a+b and Y=c+d
so X+Y=m and (m+1) possible couples are (X=0, Y=m), (X=1, Y=m-1), ..., (X=m, Y=0)
On the other hand, (a, b) such that a+b=k and a,b>=0, the number of such couples is equal to k+1.
Therefore, generic solution is:
1*(m+1)+2*m+3*(m-1)+...+m*2+(m+1)*1
To get 120 different ways, m equals to 7, or n=15, thus C.

Sure this is not the easiest way to deal with this problem![/quote]

Actually there is a formula for this
The total number of ways of dividing n identical items among r persons, each one of whom, can receive 0,1,2 or more items = n+r-1Cr-1.
Here if each nephew gets at least 2 tickets we have n-8 tickets left to distribute among 4 nephews and each one of them can receive 0,1,2..n-8 items.
Total no of ways that can happen = (n-8)+(4-1)C(4-1)
= n-5C3
n-5C3=120 for n=15 .So C[/quote]
Please I'm confused as to how you got 15 when you solved n-5C3=120
How did you arrive at 15 by solving woth that formular? Please help, I'm stuck There! Anyone hellllppp pleaseee

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Re: Mrs. Smith has been given film vouchers. Each voucher allows   [#permalink] 27 Feb 2019, 10:15

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