3) For years, ocean going ships have been using colored flags for signaling. By changing the order of the colored flags, the ships can send out different signals. If most ships carry six flags (each of different colors) how many different signals are possible if...

a) all six flags are used? b) four flags are used?

Please help, this is very confusing to me

all six are used :

number of ways = 6! = 720

four are used:
number of ways = 4! = 24

To calculate the number of different signals possible in these scenarios, we can use the concept of permutations. A permutation is an arrangement of objects in a specific order.

a) When all six flags are used, we need to determine the total number of permutations. Since there are six flags of different colors, we can calculate the number of permutations using the formula for permutations of distinct objects, which is n!.

In this case, n = 6, so we have 6! permutations.

n! = n × (n-1) × (n-2) × ... × 2 × 1

Thus, 6! = 6 × 5 × 4 × 3 × 2 × 1 = 720

Therefore, if all six flags are used, there are 720 different signals possible.

b) When only four flags are used, we still calculate the number of permutations. However, since we have fewer objects to arrange, the formula for permutations changes. We use the formula nPr = n! / (n-r)!, where n is the total number of objects and r is the number of objects being arranged.

In this case, n = 6 (total number of flags) and r = 4 (number of flags being used).

Using the formula, we have:

nPr = 6! / (6-4)!
= 6! / 2!
= (6×5×4×3×2×1) / (2×1)
= 720 / 2
= 360

Hence, if only four flags are used, there are 360 different signals possible.

So, in summary:
a) If all six flags are used, there are 720 different signals possible.
b) If four flags are used, there are 360 different signals possible.