Question 1031493: The total number of ways in which six + and four - signs can be arranged in a line such that no two - signs occur together is
a)7!/3!
b)6!*7!/3!
c)35(correct)
d)none of these
Found 2 solutions by richard1234, Edwin McCravy:
Answer by richard1234(7193)   (Show Source):
You can put this solution on YOUR website! Between any two consecutive -'s, there has to be at least one + in between.
THe number of such ways is *equal* to the number of ways we can arrange three +'s and four -'s with no restrictions: for any arrangement of 3 +'s and 4 -'s, I can append a + after the first three -'s like so:
+--+-+- becomes +-+-++-++-
And vice versa (formally, there is a bijection between length-10 strings with 6 +'s, 4 -'s, no two consecutive -'s, and length-7 strings with 3 +'s, 4 -'s).
The number of length-7 strings with 3 +'s, 4 -'s is 7C3 = 7!/(4!*3!) = 35.
Answer by Edwin McCravy(20054)  (Show Source):
You can put this solution on YOUR website!
We only need three of the + signs to keep two of the
4 - signs from coming together, like this:
- + - + - + -
Now we must insert the remaining three + signs
The arrows below point to the 5 places where we may insert
the remaining 3 + signs:
↓ ↓ ↓ ↓ ↓
- + - + - + -
The 5 places are:
Left of the 1st - sign
Immediate left of the 2nd - sign
Immediate left of the 3rd - sign
Immediate left of the 4th - sign
Right of the 4th - sign
Case 1: We put all three +++ in one of the 5 places:
That's 5 ways.
+++-+-+-+-
-++++-+-+-
-+-++++-+-
-+-+-++++-
-+-+-+-+++
Case 2: We put a pair ++ in one of the 5 places and
a single + in another
We can choose the place to put the pair ++ in 5 ways
and there remain 4 ways we can choose the place to put
the single +.
That's 5*4 = 20 ways.
Case 3: We place a single + in 3 different places.
We can choose the 3 places to put the 3 single +'s
in 5C3 = 10 ways.
Grand total: 5+20+10 = 35 ways.
Edwin
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