SOLUTION: How many permutations of the numbers 1, 2 , 3, 4, and 5 have the 4 and 5 separated by exactly a. 1 other number? b. 2 other numbers? c. 3 other numbers? for a. I tried

Question 786898: How many permutations of the numbers 1, 2 , 3, 4, and 5 have the 4 and 5 separated by exactly
a. 1 other number?
b. 2 other numbers?
c. 3 other numbers?
for a. I tried 3x2x1=6 times 2 (switch 4 and 5 around) =12 and then 12 x 3 =36 permutations.
I'm not sure if that's correct, but I really need help with part b and c because I'm stumped. I've tried and tried but no luck :(
Answer by xinxin(76) (Show Source): You can put this solution on YOUR website!
Use slot method：
a) 4 and 5 can be placed in different slots. There are 6 outcomes:
_4_ _X_ _5_ _A_ _B_, _X_ _4_ _A_ _5_ _B_, _X_ _A_ _4_ _B_ _5_
_5_ _X_ _4_ _A_ _B_, _X_ _5_ _A_ _4_ _B_, _X_ _A_ _5_ _B_ _4_
(X,A,B are chosen from 1,2,3)
For the first outcome, # of permutations = 1*3*1*2*1=6. The other five have the same number. So the total number of permutations are 6*6 =36.
Your answer is correct but I don't think you made correct explanations.
b)follow the same step as part a),
_4_ _X_ _A_ _5_ _B_, _X_ _4_ _A_ _B_ _5_
_5_ _X_ _A_ _4_ _B_, _X_ _5_ _A_ _B_ _4_
total # of permutations = 4* (1* 3*2*1*1）= 24
c) _4_ _X_ _A _ _B_ _5_, _5_ _X_ _A _ _B_ _4_
total # of permutations = 2*(1*3*2*1*1)= 12
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