A logistics firm generates a 4-character tracking code, c_1 c_2 c_3 c_4, for express parce… — GMAT

A logistics firm generates a 4-character tracking code, $c_1 c_2 c_3 c_4$ , for express parcels according to the following rules:
- The first character, $c_1$ , must be a non-zero digit ( $1$ through $9$ ).
- The second character, $c_2$ , must be a vowel chosen from the set $\{\text{A, E, I, O, U}\}$ .
- The third character, $c_3$ , must be an even digit ( $0, 2, 4, 6, 8$ ).
- The fourth character, $c_4$ , must be a consonant chosen from the set $\{\text{B, F, G, K, M, P}\}$ .

If no digit may be repeated within the code, how many different tracking codes can be created?

Answer: 1230

Answer

1230

To find the total number of unique tracking codes, calculate the independent choices for letters and dependent choices for digits. For digits

c_1

and

c_3

, if

c_1

is odd (

1, 3, 5, 7, 9

), there are

5 \times 5 = 25

ways to choose

c_3

. If

c_1

is even (

2, 4, 6, 8

), there are

4 \times 4 = 16

ways to choose

c_3

, giving

41

valid digit pairs. Multiplying by the

5

vowel options for

c_2

and

6

consonant options for

c_4

yields

41 \times 5 \times 6 = 1230

Step-by-Step Solution

Determine valid digit choices for

c_1

and

c_3

considering the non-repetition constraint

Split into Case 1 (

c_1

is odd) and Case 2 (

c_1

is even)

The availability of even digits for

c_3

depends on whether

c_1

uses an even or odd digit.

Calculate the number of ways to pick

c_1

and

c_3

Case 1 (

c_1 \in \{1, 3, 5, 7, 9\}

5 \times 5 = 25

ways. Case 2 (

c_1 \in \{2, 4, 6, 8\}

4 \times 4 = 16

ways. Total digit combinations =

25 + 16 = 41

c_1

is odd, all 5 even digits remain for

c_3

. If

c_1

is even, only 4 even digits remain for

c_3

Calculate the number of ways to pick letters

c_2

and

c_4

5 choices for

c_2

and 6 choices for

c_4

, giving

5 \times 6 = 30

letter combinations.

Vowels and consonants are chosen independently from distinct sets without restrictions between them.

Multiply independent stage outcomes using the Fundamental Counting Principle

41 \times 30 = 1230

The overall number of distinct codes is the product of digit pair combinations and letter pair combinations.

Key Concept

Fundamental Counting Principle with Conditional Restrictions

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