Goofing around
How many ways can you rearrange the letters of GOOFING so that no letter winds up in the same spot?
Clarification: The "O's" and "G's" are indistinguishable, so the resulting set of letters can't have an O in the second or third spot or a G in the first or last spot.
For more Permutations quizzes.
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The answer is 230.
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1 solution
My solution is far from elegant, but I split the problem up into 8 possibilities, and counted up the total for each one.
1) First and last letters are O:
1A: 2nd + 3rd letters are G's
1B: 2nd + 3rd letters - Only one is a G
1C: 2nd + 3rd letters - Neither is a G
2) One of the first and last letters is an O:
2A: 2nd + 3rd letters are G's
2B: 2nd + 3rd letters - Only one is a G
2C: 2nd + 3rd letters - Neither is a G
3) Neither of the first or last letters are O's:
3A: 2nd + 3rd letters are G's
3B: 2nd + 3rd letters - Only one is a G
So if N(S) = Number of ways to form state S, then:
N ( S ) = N ( 1 A ) + N ( 1 B ) + N ( 1 C ) + N ( 2 A ) + N ( 2 B ) + N ( 2 C ) + N ( 3 A ) + N ( 3 B )
N ( S ) = 2 + 6 + 1 2 + 1 8 + 9 6 + 3 6 + 1 2 + 4 8 = 2 3 0