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# Let abcd be a general four-digit number and all the digits are non-zer

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Senior Manager
Joined: 12 Jan 2019
Posts: 277
Let abcd be a general four-digit number and all the digits are non-zer  [#permalink]

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05 Apr 2019, 02:51
3
00:00

Difficulty:

75% (hard)

Question Stats:

49% (02:17) correct 51% (02:34) wrong based on 39 sessions

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1) 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
Senior Manager
Joined: 12 Jan 2019
Posts: 277
Re: Let abcd be a general four-digit number and all the digits are non-zer  [#permalink]

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05 Apr 2019, 02:51
1) We need sets of three distinct integers {a, b, c} that have a sum of one-digit number d. There are seven possibilities:

a) {1, 2, 3}, sum = 6
b) {1, 2, 4}, sum = 7
c) {1, 2, 5}, sum = 8
d) {1, 3, 4}, sum = 8
e) {1, 2, 6}, sum = 9
f) {1, 3, 5}, sum = 9
g) {2, 3, 4}, sum = 9

For each set, the sum-digit has to be in the one’s place, but the other three digits can be permutated in 3! = 6 ways in the other three digits. Thus, for each item on that list, there are six different possible four-digit numbers. The total number of possible four-digit numbers would be 7*6 = 42. Answer = (E)
Re: Let abcd be a general four-digit number and all the digits are non-zer   [#permalink] 05 Apr 2019, 02:51
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