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 ____.

Solution:

The number of one-one functions $=^{7} C_{5} \cdot 5! = 21 \cdot 120 = 2520 = α$

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

$\begin{aligned} β = \frac{7!}{3! 4!} \times 5! + \frac{7!}{2! 2! 2! 3!} \times 5! = 4 \cdot^{7} C_{3} \times 5! \\ ∴ \frac{β - α}{5!} = 4 \cdot^{7} C_{3} -^{7} C_{5} = 4 \times 35 - 21 = 119 \end{aligned}$

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 $\frac{β - α}{5!}$ is ____.

Solution:

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