Question -
Answer -
There are 7 letters inthe word ‘ARRANGE’ out of which 2 are A’s, 2 are R’s and the rest all aredistinct.
So by using theformula,
n!/ (p! × q! × r!)
total number ofarrangements = 7! / (2! 2!)
= [7×6×5×4×3×2×1] /(2! 2!)
= 7×6×5×3×2×1
= 1260
Let us consider allR’s together as one letter, there are 6 letters remaining. Out of which 2 timesA repeats and others are distinct.
So these 6 letters canbe arranged in n!/ (p! × q! × r!) = 6!/2! Ways.
The number of words inwhich all R’s come together = 6! / 2!
= [6×5×4×3×2!] / 2!
= 6×5×4×3
= 360
So, now the number ofwords 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
Hence, the totalnumber of arrangements of word ARRANGE in such a way that not all R’s cometogether is 900.