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The number of positive integral solutions of the equation a+b+c+d+e =

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The number of positive integral solutions of the equation a+b+c+d+e =  [#permalink]

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New post 31 May 2020, 01:53
2
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A
B
C
D
E

Difficulty:

  45% (medium)

Question Stats:

58% (02:05) correct 42% (01:57) wrong based on 19 sessions

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The number of positive integral solutions of the equation a+b+c+d+e = 30 is?
A.25173
B.23517
C.25731
D.23751
E.23518
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The number of positive integral solutions of the equation a+b+c+d+e =  [#permalink]

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New post 31 May 2020, 02:59
MBADream786 wrote:
The number of positive integral solutions of the equation a+b+c+d+e = 30 is?
A.25173
B.23517
C.25731
D.23751
E.23518


So 30 has to be distributed amongst 5 sets a,b,c,d and e. As we are looking at positive integral solution, let us give 1 each to all 5.
Remaining now =30-5=25
So let us now add 4 partItions so that we can distribute these in 5 sets and then choose these 4 partitions
=> (25+4)C4=29*28*27*26/4!=29*7*9*13=23751

D
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Re: The number of positive integral solutions of the equation a+b+c+d+e =  [#permalink]

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New post 31 May 2020, 05:09
MBADream786 wrote:
The number of positive integral solutions of the equation a+b+c+d+e = 30 is?
A.25173
B.23517
C.25731
D.23751
E.23518


RULE:

For any equation a+b+c+d = n

where n = Total Balls to be distributed and r = 4 (Number of variables)

The number of Positive solutions \(= (n-1)C_{r-1}\)



i.e. Positive Integer solution of the equation a+b+c+d+e = 30 will be \((30-1)C_{5-1} = 23751\)

Answer: Option D

DERIVATION OF PARTITION RULE:

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Re: The number of positive integral solutions of the equation a+b+c+d+e =   [#permalink] 31 May 2020, 05:09

The number of positive integral solutions of the equation a+b+c+d+e =

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