a) In this question we treat it as a simple permutation and hence the answer would appear to be 6! However, the two T's are indistinguishable and hence and are actually the same arrangement but appear as two separate permutations. As this happens in every single case the number of arrangements is .
b) With the T's together we treat the two T's as one letter. If we give TT the symbol , Then we are finding the permutations of , Which are arrangements.
c) For the T's separated, we remove the T's initially and find the number of permutations of LIER which is 4!
If we now consider the specific permutation REIL, then the two T's can be placed in two of five positions. This is shown in the diagram below.
Hence for the permutation REIL there are ways of positioning the T's. As this can happen with each of the 4! permutations of the four letters, then the total number of permutations is arrangements.
We could also think about this another way. As we know the total number of arrangements is 360 and the T's either have to be together or separated then the number of arrangements where they are separated is 360 minus the number of arrangements where they are together.
This gives arrangements.
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