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?
10*9*8*7
To determine the number of signals that can be sent with these flags taken four at a time, we can use the concept of combinations.
In mathematics, combinations refer to the number of ways to select a certain number of items from a larger set disregarding the order. The formula to calculate combinations is:
nCr = n! / (r! * (n - r)!)
Where n is the total number of items and r is the number of items being selected.
In this case, there are 10 different signal flags, and we want to select 4 flags at a time. Therefore, we can calculate the number of signals as follows:
nCr = 10! / (4! * (10 - 4)!)
Calculating this expression:
10! = 10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1
4! = 4 * 3 * 2 * 1
(10 - 4)! = 6 * 5 * 4 * 3 * 2 * 1
Plugging these values into the formula:
nCr = (10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1) / ((4 * 3 * 2 * 1) * (6 * 5 * 4 * 3 * 2 * 1))
Simplifying the expression:
nCr = (10 * 9 * 8 * 7) / (4 * 3 * 2 * 1)
Calculating the numerator and denominator:
nCr = 5040 / 24
Finally, dividing 5040 by 24:
nCr = 210
Therefore, the number of signals that can be sent with these flags, taken 4 at a time, is 210.