In how many ways can the letters of the word "ARRANGE" be arranged so that
(i)the two R's are never together?
(ii) the two A's are together but not the two R's ?
(iii)neither the two A's nor the two R's are together?
Answers
Answer:
No. of letters = 7
Out of these 7 letters, R repeats 2 times, and A repeats 2 times.
So, total no. of words formed by rearranging = 7!/2!2! = 5040/4 = 1260 words.
Now,
(1) The two Rs are never together:
Let us consider the 2 Rs as one letter,
then total no. of letters = 6
And no. of words formed = 6!/2! (2! for the 2 As)
= 720/2 = 360 words.
Thus, in these 360 words, both Rs remain together. Subtracting this from total no. of words = 1260 - 360 = 900 words
Hence, in 900 words, both Rs are never together.
(2)This time, let's consider both AAs as one letter.
Then, total no. of letters = 6
and words formed = 6!/2! (2! for the 2 Rs)
= 720/2 = 360 words.
Now, let us see in how many words, both Rs and both As are together.
Considering AA as one letter and RR as one letter,
no. of letters now = 5
words formed = 5! = 120
Thus, from all the words in which AAs appear together, there are 120 words in which both Rs also appear together. Therefore, we subtract this no. from the total no. of words wherein As are together.
=) 360 - 120 = 240 words.
(3) Neither the 2 As nor the 2 Rs are together:
We have already seen,
no. of words in which As are together = 360
no. of words in which Rs are together = 360
no. of words common to both the above categories = 120 (A,R both are together)
Thus, total no. of words in which either As or Rs are together = (360 + 360) - 120 = 720 - 120 = 600
Subtracting this from total no. of words = 1260 - 600 = 660