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Let X be a set with exactly 5 element...

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 function from `X` to `Y` and `beta` is the number of onto function from `Y` to `X` , then the value of `1/(5!)(beta-alpha)` is _____.

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The correct Answer is:
119
Given, X has exactly 5 elements and Y has exactly 7 elements .

`therefore n(X) =5 and n(Y) =7`
Now, number of one-one functions from X to Y is
`alpha =""^(7)P_(5)=""^(7)C_(5) xx 5!`
Number of onto functions from Y to X is `beta`
`1,1,1,1,3 or 1,1,1,2,2 `
`therefore beta = (7!)/(3!4!)xx5!+(7!)/((2!)^(3)3!)xx5!`
`=(""^(7)C_(3)+3""^(7)C_(3))5! =4xx""^(7)C_(3)xx5!`
`therefore (beta-alpha)/(5!)=((4xx""^(7)C_(3)-""^(7)C_(5))5!)/(5!)`
`=4xx35-21=140-21 =119`
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