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Zorluk: Çok zorFundamental Counting Principle

A security system requires each employee to create a 6-character access code consisting of 3 non-zero digits followed by 3 letters from the standard 26-letter English alphabet. The access code must satisfy the following restrictions:

1. The first digit must be odd.
2. No two adjacent digits in the code can be identical.
3. The first letter must be a vowel (A,E,I,O,UA, E, I, O, U), the second letter must be a consonant, and the third letter cannot be identical to the first letter.

How many distinct access codes can be created under these rules?

  1. A
    806,400806,400
  2. 840,000840,000Cevap
  3. C
    1,040,0001,040,000
  4. D
    1,063,1251,063,125
  5. E
    2,9452,945

Cevap

840,000840,000
The total number of valid codes is obtained by multiplying the number of choices at each stage. For the digits: 55 (odd non-zero) ×8\times 8 (non-zero digit ≠\neq 1st digit) ×8\times 8 (non-zero digit ≠\neq 2nd digit) =320= 320. For the letters: 55 (vowels) ×21\times 21 (consonants) ×25\times 25 (any letter ≠\neq 1st letter) =2,625= 2,625. The overall number of codes is 320×2,625=840,000320 \times 2,625 = 840,000.

Adım Adım Çözüm

1
Calculate the number of choices for the first 3 digit positions.
5 choices for the 1st digit, 8 choices for the 2nd digit, and 8 choices for the 3rd digit.
Non-zero odd digits are {1,3,5,7,9}\{1, 3, 5, 7, 9\} (5 choices). The 2nd digit can be any non-zero digit except the 1st digit (9−1=89 - 1 = 8 choices). The 3rd digit can be any non-zero digit except the 2nd digit (9−1=89 - 1 = 8 choices). Thus, 5×8×8=3205 \times 8 \times 8 = 320 digit combinations.
2
Calculate the number of choices for the 3 letter positions.
5 choices for the 1st letter, 21 choices for the 2nd letter, and 25 choices for the 3rd letter.
There are 5 vowels and 21 consonants in the 26-letter alphabet. The 1st letter has 5 choices. The 2nd letter has 21 choices (consonant). The 3rd letter can be any letter except the 1st letter (26−1=2526 - 1 = 25 choices). Thus, 5×21×25=2,6255 \times 21 \times 25 = 2,625 letter combinations.
3
Apply the Fundamental Counting Principle to find the total number of access codes.
320×2,625=840,000320 \times 2,625 = 840,000
Since choosing the digits and choosing the letters are independent stages of creating a code, multiply the total number of digit combinations by the total number of letter combinations.

Anahtar Kavram

Fundamental Counting Principle with Positional Restrictions
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Fundamental Counting Principle

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