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Difficulty: MediumCounting with Restrictions and Repetitions

A company creates 5-digit employee identification codes using all of the digits 1,1,2,2,1, 1, 2, 2, and 33. How many distinct 5-digit identification codes can be formed such that the two 22's are not adjacent to each other?

  1. A
    12
  2. 18Answer
  3. C
    24
  4. D
    30
  5. E
    60

Answer

18
To find the number of distinct 5-digit identification codes where the two 2's are not adjacent, use complementary counting. First, compute the total distinct arrangements of the digits 1,1,2,2,31, 1, 2, 2, 3, which is 5!2!×2!=30\frac{5!}{2! \times 2!} = 30. Next, find the number of arrangements where the two 2's are adjacent by treating (22)(22) as a single item. Arranging (22),1,1,3(22), 1, 1, 3 yields 4!2!=12\frac{4!}{2!} = 12 arrangements. Subtracting the adjacent arrangements from the total gives 30−12=1830 - 12 = 18.

Step-by-Step Solution

1
Calculate the total number of distinct 5-digit arrangements of the digits 1,1,2,2,31, 1, 2, 2, 3 without any restrictions.
Total arrangements = 5!2!×2!×1!=1204=30\frac{5!}{2! \times 2! \times 1!} = \frac{120}{4} = 30.
When arranging elements with repeated indistinguishable items, divide n!n! by the factorials of the counts of each repeated item.
2
Calculate the number of restricted (forbidden) arrangements where the two 22's are adjacent.
Adjacent arrangements = 4!2!×1!=242=12\frac{4!}{2! \times 1!} = \frac{24}{2} = 12.
Treat the two adjacent 22's as a single block (22)(22). We now arrange 4 items: (22),1,1,3(22), 1, 1, 3, where the digit 11 appears twice.
3
Subtract the forbidden arrangements from the total arrangements using complementary counting.
Non-adjacent arrangements = 30−12=1830 - 12 = 18.
Complementary counting gives the number of valid arrangements where the two 22's are not adjacent.

Key Concept

Permutations with Repeated Indistinguishable Elements and Complementary Counting
Estimated Time:1m 30s
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Topics

Counting with Restrictions and Repetitions

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