2
$\begingroup$

How many different strings of five characters are there if only lower-case letters or numbers can be used in creating these strings?

Here is my solution:

There are 26 letters in the alphabet and 10 numbers $\{0,1,2,3,4,5,6,7,8,9\}$, which gives us a total of 36 characters. Thus, I was able to calculate this using the following:

$$(36)^5 = 60466176$$

Now is it possible to compute the number of different strings of five characters using the following two steps?

  1. Compute the number of different strings of four characters.

$$(36)^4 = 1679616$$

  1. Create a summation that has several terms, each of them involves the number of different strings of four characters and this summation is the answer.

Here is my approach:

Let

$$X=\{0,1,2,3,4,5,6,7,8,9,a,b,c,d,e,f,g,h,i,j,k,l,m,n,o,p,q,r,s,t,u,v,w,x,y,z\}$$ Let $$N=5, M=4$$

We have $$\sum_{K=1}^{M}X^M\sum_{I=M}^{N}X^I$$

Am I on the right track? Any hints on how to approach this? I am looking to answer this question in the form of a summation.

$\endgroup$
2
  • $\begingroup$ Hint: There are $36$ characters that you can place at the trailing place of a $4$ character long string to make it a $5$ character long string. $\endgroup$
    – Eoin
    Apr 28 '15 at 5:36
  • $\begingroup$ I think I understand you. Could you please elaborate a little more? $\endgroup$
    – lucidgold
    Apr 28 '15 at 5:37
1
$\begingroup$

The number of different strings with four characters is, as your previous reasoning, $(36)^4$. Now, given a string with four characters, I can append one character to it to give me a five-character string. All five-character strings can be formed in this way ($4 + 1$), and doing this append operation gives us all five-character strings, so we know we are counting the same thing.

In how many ways can I append a character to a four-character string? $36$ ways. Since I want four characters and one extra character, I will multiply ('and' signifies multiplication, while 'or', addition). So the total number of five-character strings is $(36)^4 \cdot 36 = (36)^5$.

$\endgroup$
0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.