In how many different ways can the letters of the word ARRANGE be arranged? If the two 'R's do not occur together, then how many arrangements can be made? If besides the two R's, the two A's also do not occur together, then how many permutations will be obtained?
Answers
Answer:
660
Step-by-step explanation:
Hi,
Given word is ARRANGE which consists of 5 distinct letter with
both R and A repeating twice.
Total number of ways of arranging the letters of the word
ARRANGE are 7!/2!2! = 1260
(a)
Let X be the event in which both R's are together in the
arrangement of letters of word ARRANGE,
Lets treat R's together as one, we need to arrange 6 in 6!
because of repetition of A's twice, we need to divide by 2! ,
since the permutations between themselves result in same
arrangement.So, total number of ways X can happen are
6!/2! = 360
(b)
Let Y be the event in which both A's are together in the
arrangement of letters of word ARRANGE,
This is similar to the above scenario X,.So, total number of
ways Y can happen are 6!/2! = 360
(c)
Let Z be the event in which both R's are together and A's be
together in the arrangement of letters of word ARRANGE,
Lets treat R's together as one and A's as one, we need to
arrange 5 in 5! ways .So, total number of ways Z can happen are
5! = 120
(d)
Let W be the event in which either both R's or both A's are
together W = X ∪ Y
n(W) = n(X) + n(Y) - n(X ∩ Y)
= n(X) + n(Y) - n(Z)
= 360 + 360 - 120
= 600
Total number of ways in which either A's or R's are together are
600.
Total number of ways in which A's and R's both of them never
together are
= Total number of arrangement of word ARRANGE - Total
number of ways of arranging such that either of R's or A's are
together are
= 1260 - 600
= 660
Hope, this helps !