Number Properties Puzzle – Challenge 9

Challenge:

How many positive integers less than 10,000 are there in which the sum of the digits equals 5?

A- 31

B- 51

C- 56

D- 62

E- 93

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We are looking for a number that the sum of its digits equals 5.
Pairs possible: \(0,0,0,5; 1,4,0,0; 0,0,2,3; 1,3,1,0; 1,2,1,1; 0,1,2,2\)
Number of options for each pair:
\(0,0,0,5 = \frac{4!}{3!} = \frac{4 ×3 ×2 ×1}{3 ×2 ×1} = 4 (5, 50, 500, 5,000)\)
\(1,4,0,0 = \frac{4!}{2!} = \frac{4 ×3 ×2 ×1}{2 ×1} = 12 (14, 41, 104, 140, 410, 401, 1004, 1040, 1400, 4001, 4010, 4100)\)
\(0,0,2,3 = \frac{4!}{2!} = \frac{4 ×3 ×2 ×1}{2 ×1} = 12 (23, 32, 230, 320, …)\)
\(1,3,1,0 = \frac{4!}{2!} =\frac{4 ×3 ×2 ×1}{2 ×1)} = 12 (113, 131, 311, 1013, …)\)
\(1,2,1,1 = \frac{4!}{3!} = \frac{4 ×3 ×2 ×1}{3 ×2 ×1} = 4 (1112, 1121, 1211, 2111)\)
\(0,1,2,2 = \frac{4!}{2! }=\frac{4 ×3 ×2 ×1}{2 ×1} = 12 (122, 212, 221, 1022, …)\)
Total: \(12 + 12 + 12 + 12 + 4 + 4 = 56\)

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