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In how many ways can the letters of the word ‘ARRANGE’ be arranged so that the two R’s are never together? |
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Answer» Here's 7 letters in the word ‘ARRANGE’ out of which 2 are A’s, 2 are R’s and the rest all are distinct. Therefore by using the formula, n!/ (p! × q! × r!) The total number of arrangements = 7! / (2! 2!) = [7 × 6 × 5 × 4 × 3 × 2 × 1] / (2! 2!) = 7 × 6 × 5 × 3 × 2 × 1 = 1260 Now, let us consider all R’s together as one letter, there are 6 letters remaining. Out of which 2 times A repeats and others are distinct. Therefore these 6 letters can be arranged in n!/ (p! × q! × r!) = 6!/2! Ways. Number of words in which all R’s come together = 6! / 2! = [6 × 5 × 4 × 3 × 2!] / 2! = 6 × 5 × 4 × 3 = 360 Therefore, now the number of words in which all L’s do not come together = total number of arrangements – The number of words in which all L’s come together = 1260 – 360 = 900 Thus, the total number of arrangements of word ARRANGE in such a way that not all R’s come together is 900. |
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