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 $\alpha$ is the number of one-one functions from X to Y and $\beta$ is the number of onto functions from Y to X, then the value of $\frac{\beta -\alpha }{5!}$ is ____.

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### Solution:

The number of one-one functions${=}^{7}{C}_{5}\cdot 5!=21\cdot 120=2520=\alpha$

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

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