A standard six-sided die is rolled 6 times. You are told that among the rolls, there was one 1, 2 two's, two three 's, and one four. How many possible sequences of rolls could there have been?
permutations of {1, 2, 2, 3, 3, 4}==6!/(2!)(2!)==180 such sequences as follows:
{{1, 2, 2, 3, 3, 4}, {1, 2, 2, 3, 4, 3}, {1, 2, 2, 4, 3, 3}, {1, 2, 3, 2, 3, 4}, {1, 2, 3, 2, 4, 3}, {1, 2, 3, 3, 2, 4}, {1, 2, 3, 3, 4, 2}, {1, 2, 3, 4, 2, 3}, {1, 2, 3, 4, 3, 2}, {1, 2, 4, 2, 3, 3}, {1, 2, 4, 3, 2, 3}, {1, 2, 4, 3, 3, 2}, {1, 3, 2, 2, 3, 4}, {1, 3, 2, 2, 4, 3}, {1, 3, 2, 3, 2, 4}, {1, 3, 2, 3, 4, 2}, {1, 3, 2, 4, 2, 3}, {1, 3, 2, 4, 3, 2}, {1, 3, 3, 2, 2, 4}, {1, 3, 3, 2, 4, 2}, {1, 3, 3, 4, 2, 2}, {1, 3, 4, 2, 2, 3}, {1, 3, 4, 2, 3, 2}, {1, 3, 4, 3, 2, 2}, {1, 4, 2, 2, 3, 3}, {1, 4, 2, 3, 2, 3}, {1, 4, 2, 3, 3, 2}, {1, 4, 3, 2, 2, 3}, {1, 4, 3, 2, 3, 2}, {1, 4, 3, 3, 2, 2}, {2, 1, 2, 3, 3, 4}, {2, 1, 2, 3, 4, 3}, {2, 1, 2, 4, 3, 3}, {2, 1, 3, 2, 3, 4}, {2, 1, 3, 2, 4, 3}, {2, 1, 3, 3, 2, 4}, {2, 1, 3, 3, 4, 2}, {2, 1, 3, 4, 2, 3}, {2, 1, 3, 4, 3, 2}, {2, 1, 4, 2, 3, 3}, {2, 1, 4, 3, 2, 3}, {2, 1, 4, 3, 3, 2}, {2, 2, 1, 3, 3, 4}, {2, 2, 1, 3, 4, 3}, {2, 2, 1, 4, 3, 3}, {2, 2, 3, 1, 3, 4}, {2, 2, 3, 1, 4, 3}, {2, 2, 3, 3, 1, 4}, {2, 2, 3, 3, 4, 1}, {2, 2, 3, 4, 1, 3}, {2, 2, 3, 4, 3, 1}, {2, 2, 4, 1, 3, 3}, {2, 2, 4, 3, 1, 3}, {2, 2, 4, 3, 3, 1}, {2, 3, 1, 2, 3, 4}, {2, 3, 1, 2, 4, 3}, {2, 3, 1, 3, 2, 4}, {2, 3, 1, 3, 4, 2}, {2, 3, 1, 4, 2, 3}, {2, 3, 1, 4, 3, 2}, {2, 3, 2, 1, 3, 4}, {2, 3, 2, 1, 4, 3}, {2, 3, 2, 3, 1, 4}, {2, 3, 2, 3, 4, 1}, {2, 3, 2, 4, 1, 3}, {2, 3, 2, 4, 3, 1}, {2, 3, 3, 1, 2, 4}, {2, 3, 3, 1, 4, 2}, {2, 3, 3, 2, 1, 4}, {2, 3, 3, 2, 4, 1}, {2, 3, 3, 4, 1, 2}, {2, 3, 3, 4, 2, 1}, {2, 3, 4, 1, 2, 3}, {2, 3, 4, 1, 3, 2}, {2, 3, 4, 2, 1, 3}, {2, 3, 4, 2, 3, 1}, {2, 3, 4, 3, 1, 2}, {2, 3, 4, 3, 2, 1}, {2, 4, 1, 2, 3, 3}, {2, 4, 1, 3, 2, 3}, {2, 4, 1, 3, 3, 2}, {2, 4, 2, 1, 3, 3}, {2, 4, 2, 3, 1, 3}, {2, 4, 2, 3, 3, 1}, {2, 4, 3, 1, 2, 3}, {2, 4, 3, 1, 3, 2}, {2, 4, 3, 2, 1, 3}, {2, 4, 3, 2, 3, 1}, {2, 4, 3, 3, 1, 2}, {2, 4, 3, 3, 2, 1}, {3, 1, 2, 2, 3, 4}, {3, 1, 2, 2, 4, 3}, {3, 1, 2, 3, 2, 4}, {3, 1, 2, 3, 4, 2}, {3, 1, 2, 4, 2, 3}, {3, 1, 2, 4, 3, 2}, {3, 1, 3, 2, 2, 4}, {3, 1, 3, 2, 4, 2}, {3, 1, 3, 4, 2, 2}, {3, 1, 4, 2, 2, 3}, {3, 1, 4, 2, 3, 2}, {3, 1, 4, 3, 2, 2}, {3, 2, 1, 2, 3, 4}, {3, 2, 1, 2, 4, 3}, {3, 2, 1, 3, 2, 4}, {3, 2, 1, 3, 4, 2}, {3, 2, 1, 4, 2, 3}, {3, 2, 1, 4, 3, 2}, {3, 2, 2, 1, 3, 4}, {3, 2, 2, 1, 4, 3}, {3, 2, 2, 3, 1, 4}, {3, 2, 2, 3, 4, 1}, {3, 2, 2, 4, 1, 3}, {3, 2, 2, 4, 3, 1}, {3, 2, 3, 1, 2, 4}, {3, 2, 3, 1, 4, 2}, {3, 2, 3, 2, 1, 4}, {3, 2, 3, 2, 4, 1}, {3, 2, 3, 4, 1, 2}, {3, 2, 3, 4, 2, 1}, {3, 2, 4, 1, 2, 3}, {3, 2, 4, 1, 3, 2}, {3, 2, 4, 2, 1, 3}, {3, 2, 4, 2, 3, 1}, {3, 2, 4, 3, 1, 2}, {3, 2, 4, 3, 2, 1}, {3, 3, 1, 2, 2, 4}, {3, 3, 1, 2, 4, 2}, {3, 3, 1, 4, 2, 2}, {3, 3, 2, 1, 2, 4}, {3, 3, 2, 1, 4, 2}, {3, 3, 2, 2, 1, 4}, {3, 3, 2, 2, 4, 1}, {3, 3, 2, 4, 1, 2}, {3, 3, 2, 4, 2, 1}, {3, 3, 4, 1, 2, 2}, {3, 3, 4, 2, 1, 2}, {3, 3, 4, 2, 2, 1}, {3, 4, 1, 2, 2, 3}, {3, 4, 1, 2, 3, 2}, {3, 4, 1, 3, 2, 2}, {3, 4, 2, 1, 2, 3}, {3, 4, 2, 1, 3, 2}, {3, 4, 2, 2, 1, 3}, {3, 4, 2, 2, 3, 1}, {3, 4, 2, 3, 1, 2}, {3, 4, 2, 3, 2, 1}, {3, 4, 3, 1, 2, 2}, {3, 4, 3, 2, 1, 2}, {3, 4, 3, 2, 2, 1}, {4, 1, 2, 2, 3, 3}, {4, 1, 2, 3, 2, 3}, {4, 1, 2, 3, 3, 2}, {4, 1, 3, 2, 2, 3}, {4, 1, 3, 2, 3, 2}, {4, 1, 3, 3, 2, 2}, {4, 2, 1, 2, 3, 3}, {4, 2, 1, 3, 2, 3}, {4, 2, 1, 3, 3, 2}, {4, 2, 2, 1, 3, 3}, {4, 2, 2, 3, 1, 3}, {4, 2, 2, 3, 3, 1}, {4, 2, 3, 1, 2, 3}, {4, 2, 3, 1, 3, 2}, {4, 2, 3, 2, 1, 3}, {4, 2, 3, 2, 3, 1}, {4, 2, 3, 3, 1, 2}, {4, 2, 3, 3, 2, 1}, {4, 3, 1, 2, 2, 3}, {4, 3, 1, 2, 3, 2}, {4, 3, 1, 3, 2, 2}, {4, 3, 2, 1, 2, 3}, {4, 3, 2, 1, 3, 2}, {4, 3, 2, 2, 1, 3}, {4, 3, 2, 2, 3, 1}, {4, 3, 2, 3, 1, 2}, {4, 3, 2, 3, 2, 1}, {4, 3, 3, 1, 2, 2}, {4, 3, 3, 2, 1, 2}, {4, 3, 3, 2, 2, 1}}==180 permutations.
+118667 Very impressive guest.
Let's see if I can find a quicker way...
1,2,2,3,3,4
A standard six-sided die is rolled 6 times.
You are told that among the rolls, there was one 1, 2 two's, two three 's, and one four.
How many possible sequences of rolls could there have been?
6!/(2!2!) = (2*3*4*5*6)/4 = 2*3*5*6 = 180 ways
or
How many ways to place the 2's 6C2 = 15
for each of those there are 4C2 ways to place the 3's = 6
then 2 ways to place the 4
and then only 1 way to place the 1
15*6*2=180 ways
So we have done it 3 ways in total and each time we got the answer 180 ways 
