Question

animali 32. How many strings of six lowercase letters from the En- glish alphabet contain IS a) the letter a? b) the letters a and b? Osadno-o lls i d c) the letters a and b in consecutive positions with a preceding b, with all the letters distinct?) d) the letters a and b, where a is somewhere to the left of b in the string, with all the letters distinct? into

Answer 1

First of all to be able to do these types of problems you need to develop an approach to how to tackle these problems. Let's discuss that approach and side by side we'll answer all these problems.

(a) In first problem, you need to find the number of strings that contain character 'a' while there is no other constraint. This problem can be easily done by finding total number of strings and then subtracting those strings which don't have a. For total number of strings each space can be filled with 26 available alphabets, and for strings without 'a' each string space can be filled with 25 available alphabets. Therefore, answer is (26)⁶-(25)⁶.

(b) In order to find the words that contain alphabets a and b, we can use the same approach.Required answer= Total number of strings - (Strings with no 'a' + Strings with no 'b') + Strings with no 'a' & no 'b'.

Answer: (26)⁶-(25)⁶-(25)⁶+(24)⁶.

(c) Now, since, a and b are required to be arranged in consecutive positions with a preceding b thus, we can think of them as one unit only (ab). Thus, now to fill the spaces with the unit (ab) we need to select 2 consecutive spaces out of 6 which can be selected in 5 ways only ((1,2),(2,3),(3,4),(4,5),(5,6)).

1

2

3

4

5

6

Now, for rest of the 4 available places, since all the letters should be different and we have already used a and b so, we are left with 24 letters. So, select 4 letters out of 24 and arrange them in their respective places. Final answer can be written as 5X X 4!.

(d) Now, we have 2 fixed letters a and b and we have to select 4 letters out of 24 and arrange all these letters in 6! ways.

But, in half of these arrangements, a will be preceded by b while in the other half b will be preceded by a. So, final answer can be written as (1/2) X X 6!