1. Determine how many strings of n lowercase letters from the English alphabet c
ID: 3753164 • Letter: 1
Question
1. Determine how many strings of n lowercase letters from the English alphabet contain
(a) the letter a.
(b) the letters a and b.
(c) the letters a and b in consecutive positions with a preceding b, with all letters of the string distinct.
(d) the letters a and b, where a is somewhere to the left of b in the string, with all letters distinct.
2. Five rooms of a house are to be painted in such a way that rooms with an interconnecting door have different colors. If there are n colors available, how many different color schemes are possible when the rooms in the house are arranged in the following way?
(a) Connected rooms form a linear order with one door interconnecting two adjacent rooms.
(b) Connected rooms form a linear order with one door interconnecting two adjacent rooms. The first and last rooms must be colored differently.
(c) Connected rooms form a circular order with one door interconnecting two adjacent rooms.
Explanation / Answer
1. Determine how many strings of n lowercase letters from the English alphabet contain
lowercase letters strings =n
(a) the letter a.
no of strings = total no of strings without restriction- strings that do not contain letter a
no of strings = 26n - 25n
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(b) the letters a and b.
Here Inclusion-Exclusion Principle will be applicable
no of strings = 26n - 2*25n + 24n
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(c) the letters a and b in consecutive positions with a preceding b, with all letters of the string distinct.
Now ab is a block and we left with 24 other letters.
the string consists of n lowercase letters, there are (n-1) positions for ab
example: abx1x2x3....xn-2, x1abx2x3...xn-2
remaining all letters are distinct n=24 and r=24
P(24,24) = 24!/0! =24!
using product rule: 24!*(n-1)
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(d) the letters a and b, where a is somewhere to the left of b in the string, with all letters distinct.
we need 2 positions out of n : P(n,2)
Exactly half will have a proceeding b = P(n,2)/2
remaining all letters are distinct n=24 and r=24
P(24,24) = 24!/0! =24!
using product rule: 24!*P(n,2)/2
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