A password consists of two alphabets from... - Solved | MCA Prep

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Question 1

Mathematics Permutation and Combination Hard

A password consists of two alphabets from English followed by three numbers chosen from 0 to 3. If repetitions are allowed, the number of different passwords is:

(A) <sup>26</sup>P<sub>1</sub> × <sup>26</sup>P<sub>2</sub> × <sup>4</sup>P<sub>1</sub> × <sup>3</sup>P<sub>1</sub> × <sup>2</sup>P<sub>1</sub>

(B) (<sup>26</sup>P<sub>1</sub>)<sup>2</sup> × (<sup>4</sup>P<sub>1</sub>)<sup>3</sup>

(C) <span style="position: relative; line-height: 0; vertical-align: baseline; top: -0.5em;font-size:10.5px;">26</span>P<span style="position: relative; line-height: 0; vertical-align: baseline; bottom: -0.25em;font-size:10.5px;">1</span> × <span style="position: relative; line-height: 0; vertical-align: baseline; top: -0.5em;font-size:10.5px;">26</span>P<span style="position: relative; line-height: 0; vertical-align: baseline; bottom: -0.25em;font-size:10.5px;">2</span> × <span style="position: relative; line-height: 0; vertical-align: baseline; top: -0.5em;font-size:10.5px;">4</span>P<span style="position: relative; line-height: 0; vertical-align: baseline; bottom: -0.25em;font-size:10.5px;">1</span> × <span style="position: relative; line-height: 0; vertical-align: baseline; top: -0.5em;font-size:10.5px;">4</span>P<span style="position: relative; line-height: 0; vertical-align: baseline; bottom: -0.25em;font-size:10.5px;">2</span> × <span style="position: relative; line-height: 0; vertical-align: baseline; top: -0.5em;font-size:10.5px;">4</span>P<span style="position: relative; line-height: 0; vertical-align: baseline; bottom: -0.25em;font-size:10.5px;">3</span>

(D) (<span style="position: relative; line-height: 0; vertical-align: baseline; top: -0.5em;font-size:10.5px;">26</span>P<span style="position: relative; line-height: 0; vertical-align: baseline; bottom: -0.25em;font-size:10.5px;">1</span> × <span style="position: relative; line-height: 0; vertical-align: baseline; top: -0.5em;font-size:10.5px;">4</span>P<span style="position: relative; line-height: 0; vertical-align: baseline; bottom: -0.25em;font-size:10.5px;">1</span>)<span style="position: relative; line-height: 0; vertical-align: baseline; top: -0.5em;font-size:10.5px;">2</span>

✓ Correct Answer: Option B Verified Solution

📘 Step-by-Step Solution & Detailed Explanation:

We are given that a password consists of: \[\] Two alphabets (from A to Z) \[\] Three numbers (chosen from 0 to 3) \[\] Repetitions are allowed \[\] Since repetitions are allowed, each of the two positions for alphabets can be filled with any of the 26 English letters. \[ \text{Ways to choose alphabets} = 26 \times 26 \] Each of the three positions for numbers can be filled with any of the four digits (0, 1, 2, 3). \[ \text{Ways to choose numbers} = 4 \times 4 \times 4 = 4^3 \] Since the choices for alphabets and numbers are independent, we multiply the possibilities: \[ 26^2 \times 4^3 \]

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