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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?
Solution :
Given Variables & Initial Setup
To determine the total number of 3-digit numbers that can be formed using the set of digits $S = \{1, 2, 3, 4, 5\}$, where repetition is not allowed, we define the parameters:
- Total number of available digits ($n$): $5$
- Number of positions to be filled ($r$): $3$
- Constraint: Repetition is strictly forbidden.
Step 1: Application of the Fundamental Counting Principle
A 3-digit number consists of three distinct positions: the hundreds place, the tens place, and the units place. We analyze the number of choices for each position sequentially:
| Position | Number of Available Choices | Justification |
|---|---|---|
| Hundreds Place | 5 | Any of the digits $\{1, 2, 3, 4, 5\}$ can be placed here. |
| Tens Place | 4 | Since repetition is not allowed, the digit used in the hundreds place is excluded. |
| Units Place | 3 | The digits used in the hundreds and tens places are both excluded. |
Step 2: Mathematical Calculation
By the Fundamental Counting Principle (also known as the Multiplication Principle), if one event can occur in $m$ ways and a second independent event can occur in $n$ ways, then both events can occur in $m \times n$ ways. [Per the Fundamental Counting Principle].
Total numbers = $5 \times 4 \times 3$
Total numbers = $60$
Step 3: Verification via Permutation Formula
The problem can also be expressed as finding the number of permutations of $n$ distinct objects taken $r$ at a time, denoted by $^nP_r$.
$^nP_r = \frac{n!}{(n-r)!}$
Substituting $n=5$ and $r=3$:
$^5P_3 = \frac{5!}{(5-3)!} = \frac{5!}{2!}$
$^5P_3 = \frac{5 \times 4 \times 3 \times 2 \times 1}{2 \times 1} = 5 \times 4 \times 3 = 60$ [By the definition of Permutations].
Final Solution: 60
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?
- 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?
- 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?
CBSE Solutions for Class 11 Mathematics Permutations and Combinations
Chapters in CBSE - Class 11 Mathematics
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