Probability
Classical Probability
Grade 12
Question:
<p>One mapping is selected at random from all mappings of the set \(S = \{1, 2, 3, \ldots, n\}\) into itself. If the probability that the mapping is one-one is 3/32, then the value of \(n\) is</p>
<p>(1) 2</p>
<p>(2) 3</p>
<p>(3) 4</p>
<p>(4) none of these</p>
Step-by-Step Solution
Key Concept: Total mappings from S to S is n^n (each of n elements has n choices). One-one mappings is n! (first element has n choices, second has n-1, etc.). Set up the ratio n!/n^n = 3/32 and solve for n.
<p><strong>Step 1:</strong> Find total number of mappings from S to S.</p><p>Each element in S can map to any of the n elements in S, so total mappings = n^n</p><p><strong>Step 2:</strong> Find number of one-one (injective) mappings from S to S.</p><p>For a one-one mapping: 1st element → n choices, 2nd element → (n-1) choices, ..., nth element → 1 choice. So one-one mappings = n!</p><p><strong>Step 3:</strong> Set up the probability equation.</p><p>P(one-one) = n!/n^n = 3/32</p><p><strong>Step 4:</strong> Test values of n.</p><p>For n = 2: 2!/2² = 2/4 = 1/2 ✗</p><p>For n = 3: 3!/3³ = 6/27 = 2/9 ✗</p><p>For n = 4: 4!/4⁴ = 24/256 = 3/32 ✓</p><p><strong>Step 5:</strong> Verify: 4! = 24 and 4⁴ = 256, so 24/256 = 3/32 (dividing both by 8)</p><p>∴ Answer: n = 4 (Option C)</p>
Correct Answer: C