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# Find the number of integer solutions to |a| + |b| + |c| = 10, where no

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Joined: 03 Jun 2019
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Find the number of integer solutions to |a| + |b| + |c| = 10, where no  [#permalink]

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08 Oct 2019, 17:08
11
00:00

Difficulty:

95% (hard)

Question Stats:

26% (01:58) correct 74% (02:03) wrong based on 53 sessions

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Find the number of integer solutions to |a| + |b| + |c| = 10, where none of a, b or c is 0.

A. 36

B. 72

C. 144

D. 288

E. 576

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Re: Find the number of integer solutions to |a| + |b| + |c| = 10, where no  [#permalink]

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09 Oct 2019, 10:48
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1
First, lets assume that a,b,c are all +ve to simplify the question.
The three numbers can have the following combinations:
118 --> with 3 rearrangements ( because 1 is repeated, the possibilities = $$\frac{3!}{2!} = 3$$)
127 --> with 6 rearrangements (because all numbers are different, the possibilities = $$3! = 6$$)
136 --> with 6 rearrangements
145 --> with 6 rearrangements
226 --> with 3 rearrangements
235 --> with 6 rearrangements
244 --> with 3 rearrangements
334 --> with 3 rearrangements

Total = 36
(the above part is similar to another interesting counting question mentioned here: (https://gmatclub.com/forum/in-how-many-ways-10-identical-chocolates-be-distributed-among-3-child-307384.html#p2375066)

However, each of a,b,c can be either +ve or -ve,
so the combined possibilities of the signs = 2*2*2 = 8
for details, (a,b,c) can be:
(+,+,+)
(+,+,-)
(+,-,+)
(-,+,+)
(+,-,-)
(-,+,-)
(-,-,+)
(-,-,-)
8 possibilities

so the total number of possibilities = 36*8 = 288
D
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Re: Find the number of integer solutions to |a| + |b| + |c| = 10, where no  [#permalink]

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09 Oct 2019, 13:09
|a|+|b|+|c|=10

integral solutions possible for (|a|, |b|, |c|)= 9C2=36

Now a, b and c can can be positive or negative; Hence total possible solutions for (a,b,c)= 2*2*2*36= 288

Kinshook wrote:
Find the number of integer solutions to |a| + |b| + |c| = 10, where none of a, b or c is 0.

A. 36

B. 72

C. 144

D. 288

E. 576
Intern
Joined: 30 Sep 2019
Posts: 3
Re: Find the number of integer solutions to |a| + |b| + |c| = 10, where no  [#permalink]

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13 Oct 2019, 01:33
nick1816 wrote:
|a|+|b|+|c|=10

integral solutions possible for (|a|, |b|, |c|)= 9C2=36

Now a, b and c can can be positive or negative; Hence total possible solutions for (a,b,c)= 2*2*2*36= 288

Kinshook wrote:
Find the number of integer solutions to |a| + |b| + |c| = 10, where none of a, b or c is 0.

A. 36

B. 72

C. 144

D. 288

E. 576

Hello, could you please explain why you're doing 9C2 instead of 9C3?
Manager
Joined: 20 Jul 2019
Posts: 53
Re: Find the number of integer solutions to |a| + |b| + |c| = 10, where no  [#permalink]

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13 Oct 2019, 03:35
ShreyasJavahar wrote:
nick1816 wrote:
|a|+|b|+|c|=10

integral solutions possible for (|a|, |b|, |c|)= 9C2=36

Now a, b and c can can be positive or negative; Hence total possible solutions for (a,b,c)= 2*2*2*36= 288

Kinshook wrote:
Find the number of integer solutions to |a| + |b| + |c| = 10, where none of a, b or c is 0.

A. 36

B. 72

C. 144

D. 288

E. 576

Hello, could you please explain why you're doing 9C2 instead of 9C3?

Because the formula is n-1cr-1

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Re: Find the number of integer solutions to |a| + |b| + |c| = 10, where no   [#permalink] 13 Oct 2019, 03:35
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