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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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31 May 2020, 01:53
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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
Math Expert
Joined: 02 Aug 2009
Posts: 8753
The number of positive integral solutions of the equation a+b+c+d+e =  [#permalink]

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31 May 2020, 02:59
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

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

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31 May 2020, 05:09
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 $$(30-1)C_{5-1} = 23751$$

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