Number of n length lists while maintaining original order

Question

I am trying to understand how to calculate the number of possible n-length lists from an ordered set of length k where the resultant n-length lists order must match the original ordered set.

Example: Find all length 4 lists in {A, B, C, D, E}

A B C D
A B C E
A B D E
A C D E
B C D E

5 total options.

Note that no options start with C as there is not enough ordered elements in the set remaining to populate a length 4 list.

Here's a more concrete example: https://math.stackexchange.com/questions/4905118/how-many-seven-digit-numbers-have-seven-distinct-nonzero-digits-that-appear-in-i/4905260#4905260 — Nic, Nov 22 '24 at 22:51

score 1 · Accepted Answer · answered Nov 22 '24 at 22:49

1

Note that if we choose any $k$ elements of an ordered list of size $n$, there is only one way to arrange these terms such that the order is preserved.

Can you conclude what the answer to your question is from here?

answered Nov 22 '24 at 22:49

Nic

2,296

1

Are you implying it is simply $\binom{5}4 $ ? – Mathew Sullivan Nov 23 '24 at 01:34
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Ahh! which is the same as $\binom{5}1 $ because if you take out 1 item from 5 what you are left with is 4 items with ordered preserved. Awesome. Thank you – Mathew Sullivan Nov 23 '24 at 01:41

Number of n length lists while maintaining original order

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