QJEEAdv18Ma9fd7: JEE Advanced 2018 Mathematics - Class 12 JEEAdv18Ma9fd7 Solution: Let X be a set with exactly 5 elements and Y be a set with…

Question

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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{1}{5!} (β - α)

is ..................

Accepted Answer

Given, X has exactly 5 elements and Y has exactly 7 elements.

∴

n(X) = 5

and n(Y) = 7

Now, number of one-one functions from X to Y is

α =^{7} P_{5} =^{7} C_{5} \times 5!

Number of onto functions from Y to X is

β

1, 1, 1, 1, 3 or 1, 1, 1, 2, 2

∴

β = \frac{7!}{3! 4!} \times 5! + \frac{7!}{{(2!)}^{3} 3!} \times 5!

= (^{7} C_{3} + 3^{7} C_{3}) 5! = 4 \times^{7} C_{3} \times 5!

∴

\frac{β - α}{5!} = \frac{(4 \times^{7} C_{3} -^{7} C_{5}) 5!}{5!}

= 4 \times 35 - 21 = 140 - 21 = 119

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