Question

2)The letters of the word EXCELLENT are arranged in a random order. Find the probability that:

a. the same letter occurs at each end.

b. X,C, and N occur together, in any order.

c. all 9 letters occur in alphabetical order.

Answer 1

a) First we separate out the letters as:
E - 3
X - 1
C - 1
L - 2
N - 1
T - 1

Total ways to arrange the letters is computed here as:
= Permutation of 9 letters with 3 same letters E and 2 same letters L

Now the same letter occurs at each end in the following ways:

• L at both the ends: Number of ways = Number of permutation of 7 items given 3 Es are same = 7! / 3! = 840
• E at both the ends: Number of ways = Number of permutation of 7 items given 2 Ls are same = 7! / 2! = 2520

Therefore the probability here is computed here as:

= (2520 + 840) / 30240

= 0.1111

Therefore 0.1111 is the required probability here.

b) Let XCN be together. Total ways to arrange the letters here is computed as:

= Number of permutation of 7 items out of which XCN is together and 3 similar Es, 2 similar Ls * Number of permutations of X, C and N  

Therefore the probability here is computed as:
= 2520/30240 = 0.0833

Therefore 0.0833 is the required probability here.

c) All 9 letters can occur in alphabetical ways only in 1 order. Therefore the probability here is computed as:

= 1/30240 = 0.000033

Therefore 0.000033 is the required probability here.