This problem is a base conversion problem masquerading as a "word problem". You are responsible for controlling the stage lights at a small theater. The control box has control switches for 10 separate lights, and each switch can make its corresponding light off, red, green, yellow, or blue. How many different patterns of lighting can you set up?

I tried putting 4^10, but it's incorrect, what would be the correct power?

How about 5^10?

The light being off is also part of a pattern.

OMG totally didn't see that, thank you!

You're welcome!

To solve this problem, we need to understand that each switch can be in one of 5 positions (off, red, green, yellow, or blue) independently. Since there are 10 switches, we can think of this as finding the number of possible combinations of 5 options taken 10 at a time.

To calculate this, we need to use the concept of permutations with repetitions. In this case, the formula to calculate the number of combinations is given by:

n^r

Where n is the number of options for each switch, and r is the number of switches. In this problem, n = 5 (off, red, green, yellow, blue) and r = 10.

So, the correct formula to calculate the number of different patterns of lighting is:

5^10

Calculating this, we get:

5^10 = 9,765,625

Therefore, there are 9,765,625 different patterns of lighting that you can set up in the small theater.