Mathematics (2018) | JEE 2018 | Discussion Page

Let X be a set with exactly 5 elements and Y be a set exactly 7 elements. If α is the numbers of one - one functions from X to Y and β is the number of onto functions from Y to X , then the value of $\frac{1}{5!}(\beta -\alpha)$ is ............

A) 118

B) 119

C) 121

D) 115

Answer:

Option B

Explanation:

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

n(X) =5

n(Y) =7

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

$\alpha = ^{7}P_{5} =^{7}C_{5}\times 5!$

Number of onto functions from Y to X is β

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

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

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

$\frac{\beta -\alpha}{5!}=\frac{(4\times ^{7}C_{3}-^{7}C_{5})5!}{5!}$

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

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Answer:

Explanation: