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
