Question 850586: How many different arrangements of the word ADDAX have neither the two As adjacent to each other nor the two Ds?
Answer by jim_thompson5910(35256)   (Show Source):
You can put this solution on YOUR website! Combine the D's into one "letter" to get DD. Let's call that z. So z = DD
Do the same for the A's and let w = AA
The word ADDAX would then turn into wzX
There are 3! = 3*2*1 = 6 ways to arrange these 3 letters.
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There are 5! = 5*4*3*2*1 = 120 different ways to arrange the letters in ADDAX where the A's and D's are distinct. However, we really can't tell the A's or D's apart, which is why we must divide by 2!*2! = 2*1*2*1 = 4 to get 120/4 = 30.
There are 30 ways to arrange the letters in ADDAX where the A's and D's cannot be told apart.
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In the first part, we computed the number of ways to arrange the letters where the A's and D's stick together (we got 6). In the second part, we computed the total number of ways to arrange the letters in ADDAX (we got 30). So we subtract to get
30 - 6 = 24
this means that there are 24 ways to arrange the letters in ADDAX where the two A's are NOT adjacent or the two D's are NOT adjacent.
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