Number of sub-sequences in a given sequence

Question

If I am given a sequence X = {x1,x2,....xm}, then I will have (2^m) subsequences. Can anyone please explain how can I arrive at this formula intuitivel

Answer 1

Each subsequence is defined by choosing between selecting or not selecting each of the m elements. As there are m elements, each with two possible states, you get 2^m possibilities.

Answer 2

Each element is either in a subsequence or not. Therefore, starting with the first x1 there are two sets of subsets: those with x1 included and those without. Same can be done with the smaller sub-problem {x2,...,xm}. Therefore you finally yield 2^m.

Answer 3

If you have a sequence S, what happens when you add a new element x to the end of S?

All the subsequences of S are still subsequences of your new sequence. So are all those subsequences with x added at the end.

Voilà! Every time you add an element, you double the number of subsequences.

Answer 4

Basically you will have twice as many subsequences for each new number since you'll have (2^(m-1)) "equivalent" subsequences that are shifted one space to the right (assuming horizontal ordering and adding in the right) plus the subsequence of all elements.

Answer 5

For each element in a sequence of length m, you can either select it or leave it. Thus, there are 2 ways to deal with each element. Therefore, the total no. of ways to deal with all the m elements is 2*2*2...... m times = 2^m times.

Answer 6

To anyone who is actually looking for a substring (as the title or URL might lead you to believe):

Subset: 2^n (Order doesn't matter in sets)
Subsequence: 2^n (Since we keep the original ordering, this is the same.)
Substring: n(n+1) * 1/2 (Elements must be consecutive)