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.
