Problem 11

Probability Level 2

How many five digit positive integers divisible by 3 can be formed using the digits 0, 1, 2, 3, 4, and 5 without any of the digits repeating?

Check out the set: Combinatorics .
96 625 None of the Choices 216 120

Angela Fajardo by Angela Fajardo , 19, Philippines

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1 solution

A positive integer is divisible by 3 iff the sum of its digits is also divisible by 3. Forming them from five of the digits of 0, 1, 2, 3, 4, and 5 has only two possible combinations 0 + 1 + 2 + 4 + 5 = 12 0+1+2+4+5=12 and 1 + 2 + 3 + 4 + 5 = 15 1+2+3+4+5=15 .

  • The number of five-digit integers can be formed from {1, 2, 3, 4, 5} is straightforward (= 5!).
  • The number of five-digit integers can be formed from {0, 1, 2, 4, 5} is 5! less those with 0 as the first digit (=4!).
  • Therefore, the number of five-digit integers can be formed from {0, 1, 2, 3, 4, 5} together = 5 ! + 5 ! − 4 ! = 120 + 120 − 24 = 216 =5!+5!-4! = 120+120-24 = \boxed{216} .
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