Let X be a set with exactly 5 elements and Y be a set with exactl… - Vidyadip

MCQ

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 $\frac{1}{5!}(\beta-\alpha)$ is. . . . . .

A
$120$
✓
$119$
C
$130$
D
$135$

✓

Answer

Correct option: B.

$119$

b
$\alpha={ }^7 C _5 \times 5!=21 \times 5!$

$\beta={ }^7 C _4 \times 3 \times 2 \times{ }^5 C _2 \times 3!+{ }^7 C _3 \times 5 \times 4!$

$\therefore \frac{\alpha-\beta}{5!}=119.00$

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