Suppose we are given 4 identical red flags, 2 identical blue flags, and 3 identical green flags. Find the number m of different signals that can be formed by hanging the 9 flags in a vertical line.
We can use the formula for the number of arrangements of n objects with k1 indistinguishable objects of type 1, k2 indistinguishable objects of type 2, ..., and km indistinguishable objects of type m:
n! / (k1! * k2! * ... * km!)
In this case, we have 9 flags, with 4 identical red flags (type 1), 2 identical blue flags (type 2), and 3 identical green flags (type 3). So we can substitute n=9, k1=4, k2=2, and k3=3 into the formula to get:
m = 9! / (4! * 2! * 3!)
Simplifying this expression:
m = (9 * 8 * 7 * 6) / (4 * 3 * 2 * 1) * (2 * 1) * (3 * 2 * 1)
m = 1260
Therefore, there are 1260 different signals that can be formed by hanging the 9 flags in a vertical line.