Question

How many four digit numbers \( abcd \) exist such that \( a \) is odd, \( b \) is divisible by 3, \( c \) is even and \( d \) is prime?

• A. \( 380 \)
• B. \( 360 \)
• C. \( 400 \)
• D. \( 520 \)
• E. \( 480 \)

Hint:

In counting problems, 0 is often a "trap." It is technically even and divisible by 3, but it cannot be the first digit ($a$) of a four-digit number. Since $a$ must be odd here, 0 is allowed for $b$ and $c$.

Answer 1

The Correct Option is C

Concept: We use the Fundamental Counting Principle. The total number of four-digit numbers is the product of the number of choices for each digit position.

Step 1: Count the choices for each position.

• a (odd): {1, 3, 5, 7, 9} \(\rightarrow\) 5 choices.
• b (divisible by 3): {0, 3, 6, 9} \(\rightarrow\) 4 choices. (Note: 0 is divisible by 3).
• c (even): {0, 2, 4, 6, 8} \(\rightarrow\) 5 choices.
• d (prime): {2, 3, 5, 7} \(\rightarrow\) 4 choices.

Step 2: Calculate the total product.
\[ \text{Total numbers} = 5 \times 4 \times 5 \times 4 \] \[ = 20 \times 20 \] \[ = 400 \]