Let X be a set with exactly 5 elements and Y be a set with exactly 7 elements. If α is the number of one-one functions from X to Y and β is the number of onto functions from Y to X, then the value of β-α5! is ____.

Let X be a set with exactly 5 elements and Y be a set with exactly 7 elements. If α is the number of one-one functions from X to Y and β is the number of onto functions from Y to X, then the value of β-α5! is ____.

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    Solution:

    The number of one-one functions=7C55!=21120=2520=α

    For the number of onto functions, we do division into groups as follows : 1 1 1 1 3 or 1 1 1 2 2 The number of onto functions is the sum in the two possible scenario, given by

    β=7!3!4!×5!+7!2!2!2!3!×5!=47C3×5! βα5!=47C37C5=4×3521=119

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