Strings

Let's Generalize: Let's say we have a set - $\{a_1,a_2,a_3,...,a_n\}$ .

I am asked to find the no. of strings I can make from this set( without rep.). Let us break it down:

First we find the no. of 1-element strings. Obviously there are n-such strings.

Now, we find the no. of 2-element strings.Ex.- $a_1a_2,a_2a_1,a_9a_7$ etc. You might think the answer is $\binom{n}{2}$ . Actually it is $\binom{n}{2}.2!$ . That's because you need to take account of the permutations of each chosen string.

Similarly, no. of 3- element strings is $\binom{n}{3}.3!$

and so on.

So, for an n-element set, no. of strings without repetition is: $\binom{n}{1}.1!+\binom{n}{2}.2!+\binom{n}{3}.3!+...+\binom{n}{n}.n!$ .

In our problem, $n=4$ so answer is 64.