If the number of ways of selecting n cards out of unlimited number of cards bearing the number 0,9,3, so that they cannot be used to write the number 903 is 96, then n is equal to

Options

A. 3
B. 4
C. 5
D. 6

Explanation

To solve the problem step by step, we need to determine the value of \( n \) such that the number of ways to select \( n \) cards from an unlimited supply of cards bearing the numbers 0, 9, and 3, without being able to form the number 903, equals 96.

### Step-by-Step Solution:

1. **Understanding the Problem**: We need to select \( n \) cards from the numbers 0, 9, and 3 such that the combination cannot form the number 903.

2. **Identifying Forbidden Combinations**: The combinations that would allow us to form the number 903 are:
- Using the digits in the order of 9, 0, and 3.

3. **Counting Valid Combinations**: To avoid forming the number 903, we can use the following combinations:
- Combinations that use only the digits 0 and 9 (but not in the order to form 903).
- Combinations that use only the digits 0 and 3.
- Combinations that use only the digits 9 and 3.

4. **Calculating the Number of Ways**:
- For each of the two digits chosen (e.g., 0 and 9), there are \( 2^n \) ways to select \( n \) cards because each position can either be filled with one of the two digits or left empty.
- Therefore, for the three pairs (0, 9), (0, 3), and (9, 3), the total number of ways can be calculated as:
\[ \text{Total ways} = 2^n + 2^n + 2^n = 3 \times 2^n \]

5. **Setting Up the Equation**: According to the problem, this total must equal 96:
\[ 3 \times 2^n = 96 \]

6. **Solving for \( n \)**:
- Divide both sides by 3:
\[ 2^n = \frac{96}{3} = 32 \]
- Recognize that \( 32 \) can be expressed as a power of 2:
\[ 32 = 2^5 \]
- Therefore, we have:
\[ 2^n = 2^5 \]
- This implies:
\[ n = 5 \]

### Final Answer:
Thus, the value of \( n \) is \( 5 \).

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