Find the number of unique permutations of the letters in each word. SIGNATURE RESTAURANT

Ok, we have two words:

"Signature"

The letters are: "S I G N A T U R E"

9 different letters.

Now, we can make only words with 9 letters, so we can think on 9 slots, and in each of those slots, we can input a letter of those 9.

For the first slot, we have 9 options.

For the second slot, we have 8 options (because on is already taken)

For the second slot, we have 7 options and so on.

Now, the total number of combinations is equal to the product of the number of options in each selection:

C = 9*8*7*6*5*4*3*2*1 = 362,880.

Now, our second word is Restaurant.

The letters here are " R E S T A U N" such that R, T and A appear two times each, so we have a total of 10 letters and 7 unique letters.

So first we do the same as beffore, 10 slots and we start with 10 options.

The total number of combinations will be:

C = 10*9*8*7*6*5*4*3*2*1 = 3,628,800

A lot of combinations, but we are counting only unique words.

For example, as we have two R, we are counting two times the word:

Restaurant (because we could permutate only the two letters R and get the same word)

So we must divide by two for each letter repeated.

we have 3 letters repeated, we divide 3 times by 2.

C = ( 3,628,800)/(2*2*2) = 453,600