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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?

 Jul 12, 2021
 #1
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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.

 Jul 13, 2021
 #2
avatar+114112 
+1

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    cool

 Jul 13, 2021
edited by Melody  Jul 13, 2021

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