Problem 22 · 2011 Math Kangaroo
Stretch
Counting & Probability
caseworkcareful-counting
The five-digit number \(\overline{abcde}\) is called interesting if all of its digits are different and a = b + c + d + e holds true. How many interesting numbers are there?
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Answer: C — 168
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Hint 1 of 2
The leading digit a must equal the sum of four distinct digits, so a is at least 6.
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Hint 2 of 2
For each value of a from 6 to 9, list the digit-sets, then multiply by the arrangements of the last four digits.
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Approach: case on the leading digit, then count arrangements
- Since a = b+c+d+e with four distinct digits, the smallest possible sum is 0+1+2+3 = 6, so a is 6, 7, 8 or 9.
- The sets summing to a (none equal to a) number 1, 1, 2, 3 for a = 6, 7, 8, 9: seven sets in all.
- Each set's four digits can be arranged 4! = 24 ways after the fixed leading a.
- Total = 7 × 24 = 168, choice (C).
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