There are four types of words that can be made from the letters of TOMORROW.

Type 1:  ABCD (4 distinguishable letters) each of which can be arranged in 
         4!= 24 ways; 

Type 2:  AABB (2 pairs of indistinguishable letters) each of which can be
         arranged in 4!/(2!2!) = 6 ways;

Type 3:  AAAB (3 indistinguishable letters and 1 different letter, which can be
         arranged in 4!/3! = 4 ways.

Type 4:  AABC (2 indistinguishable letters and 2 other distinguishable letters,
         each of which can be arranged in 4!/2! = 12 ways

Put these into cases:

Case 1: The ABCD-type words.  
        Choose 4 letters from {M,O,R,T,W} in 5C4 = 5 ways.
        From the above, each can be arranged in 24 ways.
        That's 5∙24=120 ways.

Case 2: The AABB-type words.
        These are the arrangements of OORR
        From the above, there are 6 of these.
        That's 6 ways for case 2. 

Case 3. The AAAB-type words.
        The three indistinguishable letters can only be O's.
        The letter of which there is only 1 is chosen from {M,R,T,W}, 4 ways.
        There are 4 ways to choose the other letter.
        From the above, each can be arranged in 4 ways.
        That's 4∙4=16 ways 

Case 4. The AABC-type words.
        Choose the letter for the two indistinguishable letters 2 ways (O or R)
        Choose the two distinguishable letters from the other 4 letters in
        4C2=6 ways
        From the above, each can be arranged in 12 ways
        That's 2∙6∙12 = 144 ways  
 
Answer: 120+6+16+144 = 286 four-letter words.

Edwin