Hint for the last Photographer Puzzle

From last Spring’s puzzles on this board, only one remains unsolved: Saturday Photographer Puzzle 1.

A photographer takes about 15 seconds to line up a group of people and take their photo. Faced with a group of 4 people, and a little extra time, he decides to take their photo in every arrangement possible (this guy always lines them up in 1 row, and never leaves gaps). How long will it take to take all the photos?

Answer again for 8 people. For 12.

Hint 1: If he has to take 4 photos, it will take 4×15 = 60 seconds, or 1 minute.
40 photos would take 40×15 = 600 seconds, 10 minutes.
And n photos would take n/4 minutes.

For Hint 2 (the big one), click ———>

Hint 2: 1 person would take 15 seconds.
2 people would take 30 seconds (AB, then BA)
3 people would not take 45 seconds. (ABC, ACB, BAC, CAB, BCA, CBA) 3 people lead to 6 photos, and 1 and a half minutes.

Why 6? Put A in the 1st spot, then there are 2 ways for B and C
Put A in the 2nd spot, then there are 2 ways for B and C.
Put A in the 3rd spot, then there are 2 ways for B and C.
That’s 3 spots for A, times 2 ways each time = 6

4 people works the same way. 4 spots for A. Each time you put A in a spot, there are still B, C, and D to place, and we already know we can place them 6 ways. So 4×6 = 24.

But we can say this better. 4x3x2x1 = 24 (4 places for A, 3 for B, 2 for C, and 1 for D. We could leave out the multiplying by 1 at the end, but it seems incomplete without it). Mathematicians use an exclamation point to show this special multiplication.

We write: 4!
We say: “Four factorial”
And we compute: 4x3x2x1 = 24

So since the photographer has 24 photos to take of the four people, it will take him 6 minutes.

If you don’t already know the answers, it might be interesting to guess how long it will take to photograph 8 and 12 people before you do the calculations