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Difficulty: MediumBasic Probability and Counting Methods

A regional sports club assigns identification codes to all of its members. Each code consists of 11 letter chosen from the set {K,L,M,N}\{K, L, M, N\}, followed by 22 digits chosen from {1,2,3,4,5}\{1, 2, 3, 4, 5\} such that no digit is repeated within a code, followed by 11 symbol chosen from {∗,#}\{*, \#\}. How many unique identification codes can be created using this system?

Answer: 160 codes

Answer

The total number of unique identification codes that can be created is 160160.
According to the Fundamental Counting Principle, to find the total number of multi-stage outcomes, multiply the number of choices at each stage. For the letter slot, there are 44 choices. For the two digit slots without repetition, there are 5×4=205 \times 4 = 20 choices. For the symbol slot, there are 22 choices. Multiplying these gives 4×20×2=1604 \times 20 \times 2 = 160 unique identification codes.

Step-by-Step Solution

1
Determine the number of available letter choices for the first slot.
4 options
The set of allowed letters {K,L,M,N}\{K, L, M, N\} contains 4 distinct elements.
2
Calculate the number of permutations for the two-digit section without repetition.
20 options
Choosing 2 distinct digits from 5 options gives 5×4=205 \times 4 = 20 possible outcomes.
3
Determine the number of available symbol choices for the last slot.
2 options
The set of allowed symbols {∗,#}\{*, \#\} contains 2 elements.
4
Multiply the number of choices for each slot using the Fundamental Counting Principle.
160 unique codes
Total codes = 4×20×2=1604 \times 20 \times 2 = 160.

Key Concept

Fundamental Counting Principle and Permutations without Repetition
Estimated Time:1m 0s
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Topics

Basic Probability and Counting Methods

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