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#### Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.  # Let abcd be a general four-digit number and all the digits are non-zer

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Magoosh GMAT Instructor G
Joined: 28 Dec 2011
Posts: 4476
Let abcd be a general four-digit number and all the digits are non-zer  [#permalink]

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3
6 00:00

Difficulty:   95% (hard)

Question Stats: 30% (02:52) correct 70% (02:39) wrong based on 146 sessions

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Let abcd be a general four-digit number and all the digits are non-zero. How many four-digits numbers abcd exist such that the four digits are all distinct and such that
a + b + c = d?

(A) 6
(B) 7
(C) 24
(D) 36
(E) 42

This is the first of a set of 15 challenging math questions. To see the whole set, as well as the OE for this question, see:
Challenging GMAT Math Practice Questions

Mike _________________
Mike McGarry
Magoosh Test Prep

Education is not the filling of a pail, but the lighting of a fire. — William Butler Yeats (1865 – 1939)
Current Student S
Joined: 14 Jul 2016
Posts: 45
Location: India
Concentration: Strategy, Marketing
Schools: AGSM '20 (A)
GMAT 1: 650 Q48 V31
GPA: 3.5
WE: Analyst (Computer Software)
Let abcd be a general four-digit number and all the digits are non-zer  [#permalink]

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2
1
mikemcgarry wrote:
Let abcd be a general four-digit number and all the digits are non-zero. How many four-digits numbers abcd exist such that the four digits are all distinct and such that
a + b + c = d?

(A) 6
(B) 7
(C) 24
(D) 36
(E) 42

This is the first of a set of 15 challenging math questions. To see the whole set, as well as the OE for this question, see:
Challenging GMAT Math Practice Questions

Mike There are 7 combinations:
abcd
1236
2349
1359
1348
1247
1258
1269
Each can be rearranged in 6 ways. Total 6*7 = 42 .
C.

Originally posted by dnalost on 27 Feb 2017, 13:39.
Last edited by dnalost on 05 Mar 2017, 21:37, edited 1 time in total.
Manager  S
Joined: 21 Jan 2016
Posts: 74
Location: India
GMAT 1: 670 Q50 V30
WE: Engineering (Computer Software)
Re: Let abcd be a general four-digit number and all the digits are non-zer  [#permalink]

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dnalost wrote:
mikemcgarry wrote:
Let abcd be a general four-digit number and all the digits are non-zero. How many four-digits numbers abcd exist such that the four digits are all distinct and such that
a + b + c = d?

(A) 6
(B) 7
(C) 24
(D) 36
(E) 42

This is the first of a set of 15 challenging math questions. To see the whole set, as well as the OE for this question, see:
Challenging GMAT Math Practice Questions

Mike There are 7 combinations:
abcd
1236
2349
1349
1348
1247
1258
1269
Each can be rearranged in 6 ways. Total 6*7 = 42 .
C.

dnalost I am only getting ways. So total of 36 ways. I guess your one 1349 is not correct one
Intern  B
Joined: 06 Oct 2015
Posts: 45
Re: Let abcd be a general four-digit number and all the digits are non-zer  [#permalink]

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Could someone help explain this to me, I'm getting 12. The way I thought about it was finding the smallest value for D possible first given a,b,and c have to be unique. This would mean D=1+2+3; D=6.

You can arrange the 1,2,and 3 six different ways (6!) so that's six cases where D=6.

The next smallest value for D is 9 (2+3+4). You can arrange the 2,3,and 4 six ways again so you are at 12 numbers.
Intern  Joined: 17 Nov 2014
Posts: 4
Re: Let abcd be a general four-digit number and all the digits are non-zer  [#permalink]

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1+2+3=6 (1236)
Arranging 1,2,3 so six combinations.

1+2+4=7 (1247) arranging 1,2,4 six combinations

1+2+5 = 8 (1258) again 6 combinations

1+2+6 = 9 (1269) again 6 combinations

1+3+4 =8 (1348) ; 6 combinations

1+3+5 =9 (1359) ; 6 combinations

2+3+4 =9 (2349); 6 combinations

Hence , total 7 × 6 = 42 combinations of such 4 digit nos.

Sent from my Moto G (4) using GMAT Club Forum mobile app
Intern  Joined: 16 Jan 2014
Posts: 3
Let abcd be a general four-digit number and all the digits are non-zer  [#permalink]

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1
if d is 3 abc will have to be (111)- >> not possible as number are distinct
if d is 4 abc will have to be 112 or some other combination with 0 -->> Both not possible as neither 0 not repetition is allowed

Like this if we see unique combinations are possible only when d is
6,7,8,9

when d = 6 abc could be (123) -> can be arranged in 3! ways so total 6 ways
when d is 7 abc could be (124) -> can be arranged in 3! ways so total 6 ways
when d is 8 abc could be
(125) -> can be arranged in 3! ways so total 6 ways
(134)-> can be arranged in 3! ways so total 6 ways
when d is 9 abc could be
126 -> arranged in 3! ways = 6 ways
135 -> arranged in 3! ways = 6 ways
234 -> arranged in 3! ways = 6 ways

If we sum all these we get 42 combinations in total
Director  P
Joined: 14 Dec 2017
Posts: 505
Location: India
Re: Let abcd be a general four-digit number and all the digits are non-zer  [#permalink]

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Given abcd is a 4 digit number, such that a,b,c,d are distinct non zero digits & a + b + c = d

Hence lets start from the max value that d can take, d = 9, hence we have {a,b,c} = {6,2,1} or {5,3,1} or {4,3,2}

each combination of {a,b,c} can be arranged among themselves in 3! = 6 ways.

Therefore for d = 9, we have 6 * 3 = 18 ways

Similarly for d =8, we have {a,b,c} = {5,2,1} or {4,3,1}, hence 12 ways

for d = 7, we have {a,b,c} = {4,2,1}, hence 6 ways

& for d = 6, we have {a,b,c} = {3,2,1}, hence 6 ways.

we cannot take d < 6, since we don't have {a,b,c} such that they satisfy a+b+c = d

Hence Total # of 4 digit numbers required = 18 + 12 + 6 + 6 = 42

Thanks,
GyM
_________________
Intern  B
Joined: 01 Mar 2017
Posts: 12
Re: Let abcd be a general four-digit number and all the digits are non-zer  [#permalink]

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Is there any way to solve it without having to list the possible digits?
For sum =6 and 7, we have 1 combination (and its arrangements), for sum= 8 we have 2 and for sum=9 we have 3.
Is that a rule to predict the number of combinations for each sum, so we can just multiply them by their arrangements of 3!, without having to figure out which they are?
mikemcgarry
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Posts: 14004
Re: Let abcd be a general four-digit number and all the digits are non-zer  [#permalink]

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_________________ Re: Let abcd be a general four-digit number and all the digits are non-zer   [#permalink] 02 Dec 2019, 14:29
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