Kate multiplied all the integers from $1$ to her age and got $1,307,674,368,000$. How old is Kate?

$\text{(A) }14\qquad\text{(B) }15\qquad\text{(C) }16\qquad\text{(D) }17\qquad\text{(E) }18$
1. We are given that Kate multiplied all the integers from $1$ to her age and got $1,307,674,368,000$. We need to find her age.
2. Let $m$ be Kate's age. Then $m! = 1,307,674,368,000$.
3. We need to determine the value of $m$ such that $m! = 1,307,674,368,000$.
4. First, we check the number of digits in $1,307,674,368,000$. It has 13 digits.
5. We know that $10! = 3,628,800$ has 7 digits.
6. We need to check factorials of numbers greater than 10 to find the one that matches $1,307,674,368,000$.

Let's calculate the factorials step-by-step:

\[
11! = 11 \times 10! = 11 \times 3,628,800 = 39,916,800
\]

\[
12! = 12 \times 11! = 12 \times 39,916,800 = 479,001,600
\]

\[
13! = 13 \times 12! = 13 \times 479,001,600 = 6,227,020,800
\]

\[
14! = 14 \times 13! = 14 \times 6,227,020,800 = 87,178,291,200
\]

\[
15! = 15 \times 14! = 15 \times 87,178,291,200 = 1,307,674,368,000
\]

7. We find that $15! = 1,307,674,368,000$.

Therefore, Kate's age is $15$.

The final answer is $\boxed{15}$.