The largest two digit prime factor of ${ }^{200}{C}_{100}$  is

If p is a prime such that $100>p>\frac{200}{3}$ , then p and 2p but not 3p belong to [1,2,….,200]. 
While just p belongs to (1,2,….,100), so p does not divide ${ }^{\text{200}}{\text{C}}_{\text{100}}$   but concider 61 the largest prime  $\frac{200}{3}$
${61}^3$  divides 200! But only 61 divides 100!.
Hence 61 divides ${ }^{\text{200}}{\text{C}}_{\text{100}}$  .