Problem 2. What is the probability of the following events when we randomly sele

Question

Problem 2. What is the probability of the following events when we randomly select a permutation of the 26 lowercase letters of the English alphabet? (a) The first 10 letters in the permutation are in alphabetical order? (b) "cat" are the first letters of the permutation and "dog" are the last? (c) letters "cat" are next to each other in the permutation? (d) "c" and "t" are separated by at least 20 letters in the permutation? (e) "c" precedes both "a" and "t in the permutation? Provide detailed justification for your answers

Explanation / Answer

a)

First 10 places are already arranged. So we have to arrange rest 16 letters in 16 places.

So number of permutations = 16!

Probability = 16!/26!

b)

First and the last 3 places are already arranged. So we have to arrange rest of the 20 places.

so number of permutations = 20!

Probability = 20!/26!

c)

Assume 'cat' as one alphabet. So we have 23+1 alphabets to arrange.

So number of permutations = 24!

Probability = 24!/26!

d)

If there are exactly 20 letters in between c and t,

Number of permutations = 24C20* 20! *4! = 24!

If there are exactly 21 letters in between c and t,

Number of permutations = 24C21* 21! *3! = 24!

If there are exactly 22 letters in between c and t,

Number of permutations = 24!

If there are exactly 23 letters in between c and t,

Number of permutations = 24!

If there are exactly 24 letters in between c and t,

Number of permutations = 24!

So total nuber of permutations = 5*24!

Probability = 5*24!/26!

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