Probability Help (die problem) A die is rolled 20 times. How many different sequences 
a) each number 1-6 is rolled exactly three times
My Answer: (20 choose 6)*(3 choose 1)
b) each number 1-6 are each rolled exactly once in the first six rolls?
My Answer: (20 choose 6)*(6 choose 1)
c) each number rolled is at least as big as the number that was rolled directly before it?
ex: 111222333344455566666
My Answer: No idea

a) I am considering it to mean atleast 3 times, which can be obtained by an exponential generating function:
\begin{align*}
  G(x) &= \left(\frac{x^3}{3!}+\frac{x^4}{4!}+\frac{x^5}{5!}\right)^6 \\
  \left[\frac{x^{20}}{20!}\right]G(x) &= \frac{11}{414720}\cdot 20! = 64530097632000
\end{align*}
b) has been answered : $$6!\, 6^{14} = 56422198149120$$
c) Use a program to generate the sequence : $$6,21,56,126,252,462,792,1287,2002,\ldots$$
Looking up in OEIS gives A000389.
Hence, for $n$ dice throws, the number of valid sequences are:
\begin{align*}
  a_n &= \binom{n+5}{5} \\
  \implies a_{20} &= \binom{25}{5} = 53130
\end{align*}
