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Q6:
Given 5 flags of different colours, how many different signals can be generated if each signal requires the use of 2 flags, one below the other?
Solution :
Given Variables & Initial Setup
The problem involves the selection and arrangement of flags to generate a signal. We are provided with the following parameters:
- Total number of distinct flags available ($n$): $5$
- Number of flags required to form one signal ($r$): $2$
- Constraint: The signal depends on the order of the flags (one below the other).
[By the Fundamental Principle of Counting, if an operation can be performed in $n$ ways and a second operation can be performed in $m$ ways, then the total number of ways to perform both operations in sequence is $n \times m$.]
Step 1: Analyzing the Selection Process
Since the signal requires two flags placed one below the other, the position matters (top position and bottom position). This is a problem of Permutation, where we calculate the arrangement of $r$ objects out of a set of $n$ distinct objects.
The number of ways to fill the positions is as follows:
- Top Position: Any of the $5$ flags can be placed here. Number of ways = $5$.
- Bottom Position: Since one flag is already used for the top position and the flags must be different, we are left with $(5 - 1) = 4$ flags. Number of ways = $4$.
Step 2: Applying the Fundamental Counting Principle
To find the total number of distinct signals ($N$), we multiply the possibilities for each position:
$N = 5 \times 4$
$N = 20$
Step 3: Verification via Permutation Formula
Alternatively, using the Permutation formula $_nP_r = \frac{n!}{(n-r)!}$:
$_5P_2 = \frac{5!}{(5-2)!}$ [Definition of Permutation]
$_5P_2 = \frac{5 \times 4 \times 3!}{3!}$
$_5P_2 = 5 \times 4 = 20$
Visual Representation of Signal Generation
Final Solution: The total number of different signals that can be generated is 20.
More Questions from Class 11 Mathematics Permutations and Combinations EXERCISE 6.1
- Q1(i): How many 3-digit numbers can be formed from the digits 1, 2, 3, 4 and 5 assuming that (i) repetition of the digits is allowed?
- Q1(ii): How many 3-digit numbers can be formed from the digits 1, 2, 3, 4 and 5 assuming that (ii) repetition of the digits is not allowed?
- Q2: How many 3-digit even numbers can be formed from the digits 1, 2, 3, 4, 5, 6 if the digits can be repeated?
- Q3: How many 4-letter code can be formed using the first 10 letters of the English alphabet, if no letter can be repeated?
- Q4: How many 5-digit telephone numbers can be constructed using the digits 0 to 9 if each number starts with 67 and no digit appears more than once?
- Q5: A coin is tossed 3 times and the outcomes are recorded. How many possible outcomes are there?
CBSE Solutions for Class 11 Mathematics Permutations and Combinations
Chapters in CBSE - Class 11 Mathematics
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