Exam

JEE Advanced

Year

2018

Paper

Paper 2

Subject

Mathematics

Chapter

Functions

JEE Advanced 2018 · Paper 2 · Mathematics · Functions

Let X be a set with exactly 5 elements and Y be a set with exactly...

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$\dfrac{1}{5 !}$\left(\beta - \alpha\right)$ is _____-.

Step-by-step solution

$n\left(X\right) = 5$
$n\left(Y\right) = 7$
$\alpha \to$ Number of one-one function $= {}^{7}{C}_{5} \times 5 !$
$\beta \to$ Number of onto function Y to X

So, $\beta = \dfrac{7 !}{3 ! 4 !} \times 5 ! + \dfrac{7 !}{{\left(2 !\right)}^{3}3 !} \times 5 ! = \left({}^{7}{C}_{3} + 3 . {}^{7}{C}_{3}\right)5 ! = 4{\times}^{7}{C}_{3} \times 5 !$
$\dfrac{\beta - \alpha}{5 !} = 4 \times {}^{7}{C}_{3} - {}^{7}{C}_{5} = 4 \times 35 - 21 = 119$