Sweet Arrival

$\dbinom{200}{100} = \frac{200!}{100! (200-100)!} = \frac{200!}{100! * 100!} =\frac{200 * 199 * ... * 100 * 99 * ... 1}{100*99*...*1 * 100*99*...*1} =\frac{200 * 199 * ... * 101}{100*99*...*1}$

Now we can start counting the factors of 3 in the numerator and denominator and cancel them out.

Looking at the denominator we have 33 numbers that are divisible by $3^1$ . $\lfloor{\frac{100}{3}} \rfloor = 33$

We use the same method for the numerator but now we subtract the factors in the range 1 ... 100.

$\lfloor{\frac{200}{3}} \rfloor - \lfloor{\frac{100}{3}} \rfloor = 33$

Both the numerator and the denominator had 33 factors of $3^1$ . We continue by looking at $3^2$

Denominator: $\lfloor{\frac{100}{3^2}} \rfloor = 11$

Numerator: $\lfloor{\frac{200}{3^2}} \rfloor - \lfloor{\frac{100}{3^2}} \rfloor = 11$

Now $3^3$

Denominator: $\lfloor{\frac{100}{3^3}} \rfloor = 3$

Numerator: $\lfloor{\frac{200}{3^3}} \rfloor - \lfloor{\frac{100}{3^3}} \rfloor = 4$

Aha! The numerator have 4 factors that are divisible by $3^3$ while the denominator only had 3. This means that the whole expression is divisible by $3^1$ .

However, we have to check $3^4$ too before we can be certain.

Denominator: $\lfloor{\frac{100}{3^4}} \rfloor = 1$

Numerator: $\lfloor{\frac{200}{3^4}} \rfloor - \lfloor{\frac{100}{3^4}} \rfloor = 1$

$3^5$ is bigger than 200 so there is no need to check that. We can conclude that the numerator has one factor of 3 more than the denominator and thus the answer is 1.

First we have to find number of 3 in each number.

Number of 3 in 100! is $\left\lfloor \frac { 100 }{ 3 } \right\rfloor +\left\lfloor \frac { 100 }{ 9 } \right\rfloor +\left\lfloor \frac { 100 }{ 27 } \right\rfloor +\left\lfloor \frac { 100 }{ 51 } \right\rfloor =33+11+3+1=48$

Number of 3 in 200! is $\left\lfloor \frac { 200 }{ 3 } \right\rfloor +\left\lfloor \frac { 200 }{ 9 } \right\rfloor +\left\lfloor \frac { 200 }{ 27 } \right\rfloor +\left\lfloor \frac { 200 }{ 81 } \right\rfloor =66+22+7+2=97$

So, the number of 3 that left in this expression is $97-2 \times 48=97-96=\boxed{1}$