Question

How many license plates can be made if the license plates contain 6 characters out of which the first two characters are distinct digits and the remaining 4 characters are distinct capital letters of the English alphabet?

• A. \(32292000\)
• B. \(2080\)
• C. \(117\)
• D. \(18962\)

Hint:

Whenever repetitions are not allowed and order matters, use permutations: \[ {}^nP_r=\frac{n!}{(n-r)!} \]

Answer 1

The Correct Option is A

Concept: Use the multiplication principle. \[ \text{Total arrangements} = (\text{Ways for digits}) \times (\text{Ways for letters}) \]

Step 1: Choose the first two distinct digits. Number of digits available \[ 10 \] Ways to arrange two distinct digits: \[\begin{aligned} {}^{10}P_2 &= 10\times9\\ &=90 \end{aligned}\]

Step 2: Choose the last four distinct capital letters. Number of letters available \[ 26 \] Ways to arrange four distinct letters: \[\begin{aligned} {}^{26}P_4 &= 26\times25\times24\times23\\ &=358800 \end{aligned}\]

Step 3: Find the total number of license plates. \[\begin{aligned} 90\times358800 &=32292000 \end{aligned}\] \[\begin{aligned} \boxed{32292000} \end{aligned}\] Hence, option \(\mathbf{(A)}\) is correct.