If for $3 \leq r \leq 30$, $\binom{30}{30-r} + 3\binom{30}{31-r} + 3\binom{30}{32-r} + \binom{30}{33-r} = \binom{m}{r}$, then $m$ equals:

Using the property $^{n}C_{r} = ^{n}C_{n-r}$, the given expression can be written as:

$^{30}C_{r} + 3 \cdot ^{30}C_{r-1} + 3 \cdot ^{30}C_{r-2} + ^{30}C_{r-3}$

This expression represents the coefficient of $x^r$ in the expansion of:

$(1+x)^{30} + 3x(1+x)^{30} + 3x^2(1+x)^{30} + x^3(1+x)^{30}$

Taking $(1+x)^{30}$ common, we get:

$(1+x)^{30}(1 + 3x + 3x^2 + x^3)$

$= (1+x)^{30}(1+x)^3$

$= (1+x)^{33}$

The coefficient of $x^r$ in $(1+x)^{33}$ is $^{33}C_{r}$.

Comparing this with $^{m}C_{r}$, we get $m = 33$.

Answer: $33$