How many different signals can be made by taking 3 different coloured flags at a time from 5 different coloured flags?

Options

A. A. 120
B. B. 60
C. C. 15
D. D. 125

Answer

To solve the problem of how many different signals can be made by taking 3 different colored flags at a time from 5 different colored flags, we can follow these steps:

### Step 1: Understand the Problem
We have 5 different colored flags and we want to select 3 flags out of these 5 to create a signal. The order in which we arrange these flags matters since different arrangements will create different signals.

### Step 2: Identify the Formula
Since the order of selection matters, we will use the permutation formula. The number of ways to arrange \( r \) items from a total of \( n \) items is given by the formula:
\[ P(n, r) = \frac{n!}{(n - r)!} \]
where \( n \) is the total number of items, \( r \) is the number of items to arrange, and \( ! \) denotes factorial.

### Step 3: Plug in the Values
In our case, \( n = 5 \) (the total number of flags) and \( r = 3 \) (the number of flags we want to arrange). Thus, we need to calculate:
\[ P(5, 3) = \frac{5!}{(5 - 3)!} = \frac{5!}{2!} \]

### Step 4: Calculate Factorials
Now, we calculate the factorials:
- \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \)
- \( 2! = 2 \times 1 = 2 \)

### Step 5: Substitute and Simplify
Now substitute the factorial values back into the permutation formula:
\[ P(5, 3) = \frac{120}{2} = 60 \]

### Step 6: Conclusion
Thus, the total number of different signals that can be made by taking 3 different colored flags at a time from 5 different colored flags is **60**.