What happens if the orderings of perms/coms are circular? i.e) people around a table; bracelets; necklaces; basically things that require objects in sequence on a round form of something.
ex) How many ways can you sit 4 people around a table?
*moving to the cafeteria to do some shananigans around the tables.
-How many things are actually moving?
ANSWER- 3 things are actually moving, which makes the equation 3!
Final answer- 3!= 6 ways
FORMULA: (n-1)!, where n= number of things to order in a circular way.
ex) Regarding this necklace, how many ways can we make this thing unique?
Thinking the same as the previous example? If so, that is incorrect. The 3D object can be flipped around, as shown in the picture. FORMULA: (n-1)!/2
ex) 5 people around a circular table
a) if person A and person B have to sit together?
{(n-1)!} *2 <--- because they can switch around (A-B, B-A) b) if person A and person B can not sit together? FORMULA: (All ways)-(undesired ways)= Answer.
4! - 3!2! = 12 ways
c) person A and person B together; person C and person D together.
2! (from the n-1 ! formula); 2! ways to organize A/B; 2! ways to organize C/D
2! * 2! *2!= 8 ways.
Ordering identical things-
ex) "How many different 'words' are possible with these letters?"
Z O O M
*thinking the answer is 4! ? 24 ways? Wrong, again!
MOZO* is the same as MO*ZO..
FORMULA: total number of ways / "total number of identicals"! = desired ways
ex) S T A T I S T I C S
10 !/ {(3!)(3!)(2!) = 50400 UNIQUE words
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