Find the Number of Positive Integral Solutions

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The number of positive integral solutions of the equation a+b+c+d+e = [#permalink] 31 May 2020, 01:53

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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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Re: The number of positive integral solutions of the equation a+b+c+d+e = [#permalink] 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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The number of positive integral solutions of the equation a+b+c+d+e = [#permalink] 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] 07 Jun 2021, 11:20

chetan2u wrote:

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

chetan2u Could you please explain what you mean by adding 4 partitions? I understood up to 30-25=5 Thanks

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The number of positive integral solutions of the equation a+b+c+d+e = [#permalink] 07 Jun 2021, 11:25

Hi Jagwik
To distribute among 5 people(suppose a,b,c,d,e are people and we are distributing 30 toffees), we need to make four partitions.One after a, one after b, one after c, and one after d, hence Four.
Hope it helps.

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Re: The number of positive integral solutions of the equation a+b+c+d+e = [#permalink] 07 Jun 2021, 11:53

Yogananda wrote:

Yogananda

I'm struggling to understand the solution. If it is not incovenient, could you please explain how it is (25+4)C4? Thanks

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Re: The number of positive integral solutions of the equation a+b+c+d+e = [#permalink] 07 Jun 2021, 17:48

Jagwik wrote:

chetan2u wrote:

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

chetan2u Could you please explain what you mean by adding 4 partitions? I understood up to 30-25=5 Thanks

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Re: The number of positive integral solutions of the equation a+b+c+d+e = [#permalink] 11 Jun 2021, 03:06

chetan2u wrote:

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

chetan2u is there any way by which we could bypass the tedious calculation and choose an answer choice thereby saving time?
I did the unit digit calculation and got stuck between Option C and D
Curious to know how you would approach it
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Re: The number of positive integral solutions of the equation a+b+c+d+e = [#permalink] 11 Jun 2021, 06:46

Jizzy wrote:

chetan2u wrote:

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

You are correct with units digit.
The other could have been to check the divisors.
Here 9 is a divisor, so the sum of digits should be divisible by 9. However, as luck would have it, the digits are exactly the same in C and D. The remaining factors 7, 13, 29 etc do not have a trick attached to it.

So, only way is to see division by 7.
25731=21000+4731=21000+4200+531=21000+4200+490+41....41 is not divisible by 7
23751=21000+2751=21000+2800-49....each term is divisible by 7
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Re: The number of positive integral solutions of the equation a+b+c+d+e = [#permalink]

11 Jun 2021, 06:46

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Find the Number of Positive Integral Solutions

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