There are two cases.

Case 1: Those 5-letter permutations that contain only one W.

They are the permutations of WINDO of which there are 5! or 120.
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Case 2. Those 5-letter permutations that contain both W's.

Choose the positions {1st,2nd,3rd,4th,5th} for the 2 W's in 5C2 = 10 ways
Choose the letter for the leftmost unchosen position 4 ways, {I,N,D, or O}
Choose the letter for the leftmost still unchosen position 3 ways.
Choose the letter for the 1 remaining unchosen position 2 ways.

That's 10*4*3*2 = 240 ways
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Grand total = 120+240 = 360

That's the answer.
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FYI, Here is an alternate way to do Case 2, where we choose the letters
instead of the positions.

Case 2. Those that contain both W's.

We choose the two W's 1 way. 
We choose the other 3 letters from {!,N,D,O}.  That's 4 choose 3 
or 4C3 = 4 ways.

Each has 2 indistinguishable W's so the number of permutations
is 5!/2! = 120/2 = 60.

That's (1)(4)(60) = 240 ways for Case 2
 
Edwin