Find the number of different signals consisting of 9 flags that can be made using 3 white, 3 red, and 3 blue.
+130477 Find the number of different signals consisting of 9 flags that can be made using 3 white, 3 red, and 3 blue.
9! / (3! * 3! * 3!) =1680
This may not be clear at first, but let me give a simple example of why it's true.
Suppose we had three white flags and two red flags.
The number of "arrangements" that we could make could make of these flags is just 5! = 120.
But note the arrangement of
R R W W W would look just like the arrangement of R R W W W where the position of the white flags at the end were interchanged in some manner. Thus, I could choose any of the three white flags to be in the 3rd position, any of the two remaining ones to be in the 4th position and the lone remaining one to be in the 5th position. And the number of arrangements of white flags at the end would just be 3 x 2 x 1 = 3!
Thus, the same thing would be true, no matter which positions the white flags occuppied. Thus, I've got to "divide away" the "repeats" of white flags (3!) in the total number of arrangements (120). The same holds true of the red flags. I have to divide away their repeats as well = 2!
Therefore, in any arrangement of n things with repeats of one (or more) indistinguishable items, I must divide the total number of arrangements, (n!), by the factorial number of each repeated thing!!!
I hope this helps........
For extra credit, how many distinguishable "words" can you make from the word "MISSISSIPPI??"
+130477 Find the number of different signals consisting of 9 flags that can be made using 3 white, 3 red, and 3 blue.
9! / (3! * 3! * 3!) =1680
This may not be clear at first, but let me give a simple example of why it's true.
Suppose we had three white flags and two red flags.
The number of "arrangements" that we could make could make of these flags is just 5! = 120.
But note the arrangement of
R R W W W would look just like the arrangement of R R W W W where the position of the white flags at the end were interchanged in some manner. Thus, I could choose any of the three white flags to be in the 3rd position, any of the two remaining ones to be in the 4th position and the lone remaining one to be in the 5th position. And the number of arrangements of white flags at the end would just be 3 x 2 x 1 = 3!
Thus, the same thing would be true, no matter which positions the white flags occuppied. Thus, I've got to "divide away" the "repeats" of white flags (3!) in the total number of arrangements (120). The same holds true of the red flags. I have to divide away their repeats as well = 2!
Therefore, in any arrangement of n things with repeats of one (or more) indistinguishable items, I must divide the total number of arrangements, (n!), by the factorial number of each repeated thing!!!
I hope this helps........
For extra credit, how many distinguishable "words" can you make from the word "MISSISSIPPI??"
