A standard six-sided die is rolled 7 times. You are told that among the rolls, there was one 1, two 2's, and three 4's. How many possible sequences of rolls could there have been? (For example, 5, 1, 2, 2, 4, 4, 4 is one possible sequence. 4, 1, 4, 4, 2, 1, 1 is one
+2668 Let's divide this problem up into cases:
Case 1 (2 1's, 2 2's, 3 4's) - There are 7!2!2!3!=210 cases.
Case 2 (1 1's, 3 2's, 3 4's) - There are 7!3!3!=140 cases.
Case 3 (1 1's, 2 2's, 4 4's) - There are 7!2!4!=105 cases.
Case 3 (1 1's, 2 2's, 3 4's, 1 3, 5, or 6) - There are 7!2!3!×3=1260 cases.
So, there are 210+140+105+1260=1,715 sequences.
