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 ____.

# 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 $\frac{\beta -\alpha }{5!}$ is ____.

Register to Get Free Mock Test and Study Material

+91

Verify OTP Code (required)

I agree to the terms and conditions and privacy policy.

### Solution:

The number of one-one functions${=}^{7}{C}_{5}\cdot 5!=21\cdot 120=2520=\alpha$

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

Register to Get Free Mock Test and Study Material

+91

Verify OTP Code (required)

I agree to the terms and conditions and privacy policy.