arixlove wrote:
Hi,
I have a doubt on permutations, because I don't understand why this 2 exercises are solved differently:
1)In how many different ways can the letters of the word MISSISSIPPI be arranged if the S's must stay together?Answer: 840 ways, solved with permutation indistinguishable items formula: 8!/2!x4!
2)In how many ways can the word ANTEDILUVIAN be arranged if the T is always in the first spot and the U and the V must stay togheter?
Ans:10!/4, still used permutat. indis. items formula,arriving to 10!/(2^3), BUT THIS TIME DUE TO THE FACT UV CAN ALSO BE ARRANGED IN 2! WAYS MULTIPLY BY IT...
What is the difference? I mean, why in the first problem we did not further multiply by 4! because the S togheter are also indistinguishable and could be arranged in 4! ways...
Thank you for the help:)
If I'm understanding your question correctly, the difference is that when you swap the S's positions with the other S's, that doesn't actually result in a new 'order,' for our purposes. So you should not multiply by 4! (...in fact, to solve this question you've *divided* those 4! arrangements into 1, already, though you might not realize this has happened).
But if you switch the U and the V, you DO get a different 'order,' for our purposes. So while you treat 'UV' like one item, for every possible placement of 'UV,' there's the reverse possibility of it being 'VU.' So you have to multiply the total by 2. If there were 3 items that had to be grouped, 'UVA,' for each possible placement of those three consecutive items, there are actually 3! = 6 different arrangements within, so you'd multiply the total by 6.
Basically, if the reordering results in a new result, you have to count each result. If swapping identical items, the results are not 'different,' so don't multiply by those possible orderings.