Permutations & Combinations
Fundamental Principle of Counting
Grade 11
Question:
<p>A variable name in certain computer language must be either a alphabet or a alphabet followed by a decimal digit. Find the total number of different variable names that can exist in that language.</p>
Step-by-Step Solution
Key Concept: Count variable names in two mutually exclusive cases: single alphabets AND alphabets followed by digits, then add them. The key is recognizing these form a partition of all valid variable names.
<p><strong>Step 1:</strong> Identify the two types of valid variable names:</p><ul><li>Type 1: Single alphabet only</li><li>Type 2: Alphabet followed by one decimal digit</li></ul><p><strong>Step 2:</strong> Count Type 1 variable names.</p><p>A single alphabet can be any letter from A-Z.</p><p>Number of Type 1 names = 26</p><p><strong>Step 3:</strong> Count Type 2 variable names.</p><p>First position: 26 choices (any alphabet A-Z)</p><p>Second position: 10 choices (any decimal digit 0-9)</p><p>By fundamental counting principle: Number of Type 2 names = 26 × 10 = 260</p><p><strong>Step 4:</strong> Find total using addition rule (mutually exclusive cases).</p><p>Total variable names = Type 1 + Type 2 = 26 + 260 = <strong>286</strong></p>
Correct Answer: 286