1. Find $x$

Let $|A|=3$ and $|B|=5$.

The number of one-one (injective) functions from $A$ to $B$ is the number of ways to assign distinct images in $B$ to the $3$ elements of $A$:

$x = {}^{5}P_{3} = 5\cdot 4\cdot 3 = 60.$

2. Find $y$

We need the number of one-one functions from $A$ to $A\times B$.

Now,

$|A\times B| = |A|\cdot |B| = 3\cdot 5 = 15.$

So the number of injective functions from a 3-element set to a 15-element set is:

$y = {}^{15}P_{3} = 15\cdot 14\cdot 13 = 2730.$

3. Compare $x$ and $y$

$\frac{y}{x} = \frac{2730}{60} = 45.5 = \frac{91}{2}.$

Thus,

$y = \frac{91}{2}x$

or equivalently,

$2y = 91x.$

4. Check options

• A: $2y=273x$ ❌
• B: $y=91x$ ❌
• C: $2y=91x$ ✅
• D: $y=273x$ ❌

Therefore, the correct option is C.