A ship carries exactly 10 different signal flags. If each possible combination and ordering of 4 of these flags connotes a specific message, how many signals can be

sent with these flags, taken 4 at a time?

If a flag can be used more than once in a signal, the number would be 10x10x10x10 = 10000

If a flag can be used only once, the number is
10x9x8x7 = 5040

To find out how many signals can be sent with these flags taken 4 at a time, we can use the concept of combinations.

Combination is a mathematical concept that calculates the number of ways to select a specified number of items from a larger set without regard to the order of the items selected.

In this case, we have 10 different signal flags and we want to select 4 flags at a time. The formula to calculate combinations is given by:

C(n, r) = n! / (r!(n-r)!),

where n is the total number of items and r is the number of items we want to select.

Applying the formula to our problem, we have:

C(10, 4) = 10! / (4!(10-4)!).

Now let's calculate it step by step:

10! = 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 = 3,628,800.
4! = 4 x 3 x 2 x 1 = 24.
(10-4)! = 6 x 5 x 4 x 3 x 2 x 1 = 720.

Substituting the values into the formula:

C(10, 4) = 3,628,800 / (24 x 720) = 210.

Therefore, there are 210 different signals that can be sent with these flags, taken 4 at a time.