Math halp pls?

Question

1. How many different ways can you arrange four friends in a row for a photograph?

2. Neon sign #1 turns on for 4 secs then off for 2 secs. Neon sign #2 turns on for 4 secs then off for 6secs. Both signs were started at the same time and repeat the cycle endlessly. how many times per minute do they start their cycles together?

Thanx
PLS so working out

Answer 1

The quick way to approach the first problem is to get 4 factorial. 4*3*2*1=24. The slow/tedius way is to show all of the arrangements.
1234
1243
1342
1324
1432
1423
2134
2143
2314
2341
2413
2431
3124
3142
3214
3241
3412
3421
4123
4132
4213
4231
4312
4321

On problem 2, the lights cycle at 6 seconds and 10 seconds.
Light 1's timing starts at 0, 6, 12, 18, 24, 30, 36, 42, 48, 54
Light 2 starts at 0, 10, 20, 30, 40, 50
( note that at 60 seconds, it is the next minute)
Therefor the lights start together at the beginning and at 30 seconds. So the answer is 2 times per minute.
A more mathematical approach is to find the product of the unique multiples of 6 and 10. (2*3 & 2*5) The unique ones are 2, 3 and 5. 2*3*5 = 30.

24 cannot be correct. It is not a multiple of the light cycling at 10 seconds (4on6off)

Answer 2

1) Think that you have 4 different ways to start the arrangement(Say on the left)
Friend 1, Friend 2, Friend 3, Friend 4...
So take 4...
Then after you pick the far left, you only have 3 options of which friend to put next(As one is already placed.
So take 3...
Then you only have 2 options for friends as the other 2 are placed.
So take 2...
Finally, there is only one left so you don't have a choice on the last friend..
Take 1...
Multiply these together 4*3*2*1=24, which is your answer.

2) Sign #1 has a 6 second cycle, and #2 has a 10 second cycle, so you just have to find any numbers between 0 and 60 which are divisible by 6 and 10.

6 12 18 24 30 36 42 48 54 60 (60 is also 0, start of new minute)

10 20 30 40 50 60 (60 is also 0)

So you can see that the ones that match are 30 and 60... So they only overlap start times twice per minute. (Every 30 seconds)

Answer 3

(1)
4*3*2*1 = 24 Possible ways to take the photograph.

(2)
For Neon sign #1, it takes 6 seconds to complete the cycle.
For Neon sign #2, it takes 10 seconds to complete the cycle.

For Neon sign #1, 6 seconds x 5 cycles = 30 seconds.
For Neon sign #2, 10 seconds x 3 cycles = 30 seconds.

The next time they start off together, it will be at the 30 second interval.
1 min./ 30 sec =
60 sec./ 30 sec. = 2
So it's two times a minute they start their cycles together.

Answer 4

When arranging or say picking from a group there ways to do them. Lets say each friend represented a number 1,2,3,4. In order to see how many different arrangements(orders) they can stand you need to figure out the factorial of it. Factorial is 4 x 3 x 2 x 1. If you had 5 friends factorial would have been 5 x 4 x 3 x 2 x 1. So for the 4 friends, you do factorial of 4 which is 24. I used an excel spreadsheet to show results for both so hope it shows here. Looks like only twice for the signs to match. Any time that the off seconds match that means they will start their cycles same as they did at beginning.

1234 sign 1 0 sign 2 0
1243 on 4 4 on 4 4
1324 off 2 6 off 6 10
1342 on 4 10 on 4 14
1423 off 2 12 off 6 20
1432 on 4 16 on 4 24
2134 off 2 18 off 6 30
2143 on 4 22 on 4 34
2314 off 2 24 off 6 40
2341 on 4 28 on 4 44
2413 off 2 30 off 6 50
2431 on 4 34 on 4 54
3124 off 2 36 off 6 60
3142 on 4 40
3214 off 2 42
3241 on 4 46
3412 off 2 48
3421 on 4 52
4123 off 2 54
4132 on 4 58
4213 off 2 60
4231
4312
4321

Answer 5

Matt's answers are good, but the answer to the second one should be 2 1/2 times per minute (5 times every two minutes). 24 seconds is correct