Problem Solving

Gmat_mission 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

Step 1: Choose 3 digits
Since 0 < d ≤ 9, a, b and c must be composed of 3 digits with a nonzero sum less than or equal to 9:
1, 2, 3 --> d = 1+2+3 = 6
1, 2, 4 --> d = 1+2+4 = 7
1, 2, 5 --> d = 1+2+5 = 8
1, 2, 6 --> d = 1+2+6 = 9
1, 3, 4 --> d = 1+3+4 = 8
1, 3, 5 --> d = 1+3+5 = 9
2, 3, 4 --> d = 2+3+4 = 9
Number of 3-digit options = 7.

Step 2: Count the number of ways each 3-digit option can be arranged
Since each option is composed of 3 distinct digits, the number of ways to arrange each 3-digit option= 3! = 6.

In counting problems:
AND means MULTIPLY.
OR means ADD.

To form a viable integer, we must choose 3 digits in Step 1 AND arrange them in Step 2.
Since we require Step 1 AND Step 2, the blue results above must be MULTIPLIED:
7*6 = 42.

The correct answer is E.