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Question:

LetXbe a set with exactly5elements andYbe a set with exactly7elements. Ifαis the number of one-one functions fromXtoYandβis the number of onto functions fromYtoX, then the value of15!(β−α)is _______.LetXbe a set with exactly5elements andYbe a set with exactly7elements. Ifαis the number of one-one functions fromXtoYandβis the number of onto functions fromYtoX, then the value of15!(β−α)is _______.XXXXX55555YYYYY77777αααααXXXXXYYYYYβββββYYYYYXXXXX15!(β−α)15!(β−α)15!(β−α)15!15!15!15!1115!5!55!!((ββ−−αα))?

Solution:

n(X)=5n(Y)=7⇒α→Number of one-one function=7C5×5!⇒β→Number of onto functionYtoX1,1,1,1,3 1,1,1,2,2⇒7!3!4!×5!+7!(2!)33!×5!=(7C3+3.7C3)5!=4×7C3×5!⇒β−α5!=4×7C3−7C5=4×35−2;1=119.