+0

# Counting

0
45
2

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?

Aug 29, 2021

#1
+1

1,  2, 2,  3,  3,  4 ==6 digits ==6! /2!.2! ==180 permutations 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)] >Total distinct permutations = 180

Aug 29, 2021
#2
+114536
0

Very impressive guest :)

Melody  Aug 30, 2021