1. We need the highest power of $3$ dividing $50!$.

   This is found using Legendre’s formula: $v_3(50!)=\left\lfloor \frac{50}{3} \right\rfloor+\left\lfloor \frac{50}{3^2} \right\rfloor+\left\lfloor \frac{50}{3^3} \right\rfloor+\left\lfloor \frac{50}{3^4} \right\rfloor+\cdots$

2. Compute each term: $\left\lfloor \frac{50}{3} \right\rfloor=16$ $\left\lfloor \frac{50}{9} \right\rfloor=5$ $\left\lfloor \frac{50}{27} \right\rfloor=1$ $\left\lfloor \frac{50}{81} \right\rfloor=0$

3. Add them: $v_3(50!)=16+5+1=22$

4. Therefore, the largest $n \in \mathbb{N}$ such that $3^n$ divides $50!$ is $n=22$

5. Checking options:

   • A: $22$ ✅
   • B: $20$ ❌
   • C: $21$ ❌
   • D: $23$ ❌

Hence, the correct option is A.