How many signals can be formed with 4 green and 3 red flags arranged vertically?

Answer

To solve the problem of how many signals can be formed with 4 green and 3 red flags arranged vertically, we can follow these steps:

### Step 1: Determine the total number of flags
We have 4 green flags and 3 red flags. Therefore, the total number of flags is:
\[ \text{Total flags} = 4 + 3 = 7 \]

### Step 2: Use the formula for permutations of multiset
The formula for the number of arrangements of \( n \) items where there are groups of indistinguishable items is given by:
\[ \frac{n!}{n_1! \times n_2!} \]
where \( n \) is the total number of items, and \( n_1, n_2, \ldots \) are the counts of indistinguishable items.

In our case, \( n = 7 \) (total flags), \( n_1 = 4 \) (green flags), and \( n_2 = 3 \) (red flags). Thus, we can write:
\[ \text{Number of arrangements} = \frac{7!}{4! \times 3!} \]

### Step 3: Calculate the factorials
Now we need to calculate the factorials:
- \( 7! = 7 \times 6 \times 5 \times 4! \) (we will keep \( 4! \) for cancellation)
- \( 4! = 24 \)
- \( 3! = 6 \)

### Step 4: Substitute the values into the formula
Substituting the values we calculated:
\[ \text{Number of arrangements} = \frac{7 \times 6 \times 5 \times 4!}{4! \times 3!} \]
Cancelling \( 4! \) from the numerator and denominator, we get:
\[ = \frac{7 \times 6 \times 5}{3!} \]

### Step 5: Calculate \( 3! \)
We already know \( 3! = 6 \), so we can substitute this into our equation:
\[ = \frac{7 \times 6 \times 5}{6} \]

### Step 6: Simplify the expression
Now we can simplify:
\[ = 7 \times 5 = 35 \]

### Conclusion
Thus, the total number of signals that can be formed with 4 green and 3 red flags arranged vertically is:
\[ \boxed{35} \]