Planck's constant $h$, speed of light $c$ and gravitational constant $G$ are used to form a unit of length $L$ and a unit of mass $M$. Then the correct option(s) is(are)

Step 1: Write the dimensional formulas of the given constants.

The dimensions of Planck's constant $h$, speed of light $c$, and gravitational constant $G$ in terms of mass ($M$), length ($L$), and time ($T$) are:

$$[h] = M L^{2} T^{-1}$$ $$[c] = L T^{-1}$$ $$[G] = M^{-1} L^{3} T^{-2}$$

Step 2: Express the unit of length $L$ in terms of $h$, $c$, and $G$.

Assume a power-law relationship for length.

$$L = h^{a} c^{b} G^{d}$$

Substitute the dimensional formulas into the relation:

$$M^{0} L^{1} T^{0} = (M L^{2} T^{-1})^{a} (L T^{-1})^{b} (M^{-1} L^{3} T^{-2})^{d}$$ $$M^{0} L^{1} T^{0} = M^{a-d} L^{2a+b+3d} T^{-a-b-2d}$$

Step 3: Solve for the exponents $a$, $b$, and $d$.

Equate the exponents of $M$, $L$, and $T$ on both sides.

$$a - d = 0$$ $$2a + b + 3d = 1$$ $$-a - b - 2d = 0$$

From the first equation:

$$a = d$$

Substitute $a = d$ into the third equation:

$$b = -3a$$

Substitute $d = a$ and $b = -3a$ into the second equation:

$$2a - 3a + 3a = 1$$ $$2a = 1$$ $$a = d = \frac{1}{2}$$ $$b = -\frac{3}{2}$$

Step 4: Determine the proportionalities for length $L$.

Substitute the exponents back into the expression for $L$.

$$L = h^{1/2} c^{-3/2} G^{1/2}$$ $$L = \frac{\sqrt{hG}}{c^{3/2}}$$

This yields the following proportionalities:

$$L \propto \sqrt{h}$$ $$L \propto \sqrt{G}$$

Step 5: Express the unit of mass $M$ in terms of $h$, $c$, and $G$.

Assume a power-law relationship for mass.

$$M = h^{p} c^{q} G^{r}$$

Substitute the dimensional formulas into the relation:

$$M^{1} L^{0} T^{0} = (M L^{2} T^{-1})^{p} (L T^{-1})^{q} (M^{-1} L^{3} T^{-2})^{r}$$ $$M^{1} L^{0} T^{0} = M^{p-r} L^{2p+q+3r} T^{-p-q-2r}$$

Step 6: Solve for the exponents $p$, $q$, and $r$.

Equate the exponents of $M$, $L$, and $T$ on both sides.

$$p - r = 1$$ $$2p + q + 3r = 0$$ $$-p - q - 2r = 0$$

From the first equation:

$$p = r + 1$$

From the third equation:

$$q = -p - 2r$$ $$q = -3r - 1$$

Substitute $p$ and $q$ into the second equation:

$$2(r + 1) + (-3r - 1) + 3r = 0$$ $$2r + 1 = 0$$ $$r = -\frac{1}{2}$$

Using $r = -\frac{1}{2}$, find $p$ and $q$:

$$p = \frac{1}{2}$$ $$q = \frac{1}{2}$$

Step 7: Determine the proportionalities for mass $M$.

Substitute the exponents back into the expression for $M$.

$$M = h^{1/2} c^{1/2} G^{-1/2}$$ $$M = \frac{\sqrt{hc}}{\sqrt{G}}$$

This yields the following proportionalities:

$$M \propto \sqrt{c}$$ $$M \propto G^{-1/2}$$

Answer: The correct options are (1), (3), and (4).