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 ___

Book Online Demo

Check Your IQ

Try Test

Courses

Dropper NEET Course Dropper JEE Course Class - 12 NEET Course Class - 12 JEE Course Class - 11 NEET Course Class - 11 JEE Course Class - 10 Foundation NEET Course Class - 10 Foundation JEE Course Class - 10 CBSE Course Class - 9 Foundation NEET Course Class - 9 Foundation JEE Course Class -9 CBSE Course Class - 8 CBSE Course Class - 7 CBSE Course Class - 6 CBSE Course

Offline Centres

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 ____.

see full answer

Start JEE / NEET / Foundation preparation at rupees 99/day !!

21% of IItians & 23% of AIIMS delhi doctors are from Sri Chaitanya institute !!

An Intiative by Sri Chaitanya

(Unlock A.I Detailed Solution for FREE)

Ready to Test Your Skills?

Check your Performance Today with our Free Mock Test used by Toppers!

Take Free Test

Detailed 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

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