<p>Let \(X\) be a set with exactly 5 elements and \(Y\) be a set with exactly 7 elements. If \(A\) is the number of one-one functions from \(X\) to \(Y\) and \(B\) is the number of onto functions from \(Y\) to \(X\), then the value of \(\dfrac{1}{5!}(B - A)\) is ________.</p>
Step-by-Step Solution
Key Concept: One-one functions from X to Y = P(7,5) = 7×6×5×4×3, and onto functions from Y to X requires Stirling numbers of second kind: B = S(7,5)×5!. The key is recognizing that onto functions count surjective mappings, which relates to distributing 7 distinct elements into 5 non-empty distinct boxes.
<p><strong>Step 1: Calculate A (one-one functions from X to Y)</strong></p><p>A one-one function from a 5-element set to a 7-element set requires selecting and arranging 5 distinct elements from 7 available positions.</p><p>A = P(7,5) = 7!/(7-5)! = 7!/2! = 7×6×5×4×3 = 2520</p><p><strong>Step 2: Calculate B (onto functions from Y to X)</strong></p><p>For onto functions from Y (7 elements) to X (5 elements), every element of X must be mapped to at least once. Using inclusion-exclusion:</p><p>B = Σ(k=0 to 4) (-1)^k × C(5,k) × (5-k)^7</p><p>B = C(5,0)×5^7 - C(5,1)×4^7 + C(5,2)×3^7 - C(5,3)×2^7 + C(5,4)×1^7</p><p>B = 1×78125 - 5×16384 + 10×2187 - 10×128 + 5×1</p><p>B = 78125 - 81920 + 21870 - 1280 + 5 = 16800</p><p><strong>Step 3: Calculate (B - A)</strong></p><p>B - A = 16800 - 2520 = 14280</p><p><strong>Step 4: Divide by 5!</strong></p><p>(B - A)/(5!) = 14280/120 = 119</p><p>∴ Answer: <strong>119</strong></p>
Correct Answer: 119