Question

[02] For n ≥ 1, how many strings of length n using letters a,b,c are there if the letter a must occur an even number of times?

Answer 1

Answer:

Given Data

For ,

How many strings of length n using letters a,b,c are there if the letter a must occur an even numberof times.

, letters a, b, c

Condition a must occurs an even no. of times

(0,2,4,6,------)times

Let the string of length is n

1) only used b,c Total length n

Each place we have two options b,c

Total string = 2n

2) Now add a,b,c to make string

a occurs two times

a occurs four times

In general

Total String of length n T(n) =

where  

Let n = 1 ,

Strings = { b , c }

Strings = 2

Using formula ,

  

  

value of k is possible

T(n) = 2n

= 21

= 2

Let n = 2 ,

Strings = {aa, bc, cb, bb, cc}

   Strings = 5

Using formula  

  

= 1

   

  

= 4+1

= 5

Let n = 3 ,

Strings = {aab, bcb, baa, aac, aca, caa, bbb, ccc, bbc, bcb, cbb, ccb, cbc, bcc}

   Strings = 14

Using formula ,

= 1

  

  

= 8 + 6

= 14

The approach of finding the total number of strings of length n using letters a,b, c are there if the letter a must occur an even number of times is:-

First finding total no of strings of length n using only b and c because there are no restrictions on b and c that is equal to 2n.

Then finding total no of strings of length n using a,b,c by considering an iteratively 2 times 4 times .........

until it will less than n.