Eight people are to make photo, four of them being standing

Stolyar,

What about,

4568
1237

...I think you missed some...

the official answer is 14. I can not comment on this.
Let's think together.

THe only way I can think to get 14 is all four in front are the shortest and must line up in one way- Left to right tallest to shortest. Thus in back we are not given the heights, but the tallest has to be the furtherest left. That leaves the other three spots to be filled. Since heights are not given= 4c3+4c2+4c1=14

I think your answer is more coincidental than explanatory. Nice try, though.

I don't know if there is a formula, so I will try to solve this logically:

First, let's say the formation looks like this:

0000
0000

where the heights go from 1 to 8 and the unknowns are 0's.

We know that at a minimum, it must look like this:

0008
1000

Now, consider the person top row 2nd from left. He must have one person shorter than him both to his left and below. So at a minimum, he must be a 4. Similarly, the person in the top row, 3rd from the left must have 5 shorter than him (two on his left, one below plus the two to the left of the one below) so his minimum height is 6.

Now we can count. Let's designate the arrangement as:

XYZ8
1000

Y can only be 4,5, or 6.

If Y is 4, Z can be 6 or 7 and X can be 2 or 3. 4 ways
If Y is 5, Z can be 6 or 7 and X can be 2, 3, or 4. 6 ways.
If Y is 6, Z must be 7 and X can be 2, 3, 4, or 5. 4 ways.

For each of the combinations above, the bottom row is determined (we have 3 numbers left all of which must be in order).

Hence, there are 4 + 6 + 4 = 14 ways.

KL, is there a better way to solve this?