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Maths Question---Possibilities

發問:

The letters of the word “SENTENCE” are rearranged in a row. (a) How many arrangements can be made?(b) If the letters are rearranged at random, how many arrangement can be made if(i) all E’s are not separated?(ii) The two N’s are separated?(iii) The arrangement begins and ends with the same letter?

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1) Totally there are 8 letters in which 2 are of the same kind (letter N) and 3 are of the another same kind (letter E). So no. of arrangements = 8!/(2! x 3!) = 3360 b i) If all E's are not separated, we can treat it as 6 entities in which "EEE" is treated as one single entity. So now we are arranging 6 entities in which 2 are of the same kind, with no. of possible arrangement = 6!/2! = 360 ii) If all N's are not separeted, we can treat it as 7 entities in which "NN" is treated as one single entity. So now we are arranging 7 entities in which 3 are of the same kind, with no. of possible arrangement = 6!/3! = 840 So if the arrangement is such that the N's are separated, no. of possible arrangements = 3360 - 840 = 2520 iii) Beginning and ending with "E": Equivalent to arranging 6 letters in which 2 are of the same kind = 6!/2! = 360 Beginning and ending with "N": Equivalent to arranging 6 letters in which 3 are of the same kind = 6!/3! = 120 So total for this part = 480

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