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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?
Solution :
Step 1: Analyzing the Positional Constraints
We are required to form a 3-digit number using the set of digits $S = \{1, 2, 3, 4, 5\}$. A 3-digit number consists of three distinct positions: the Hundreds place, the Tens place, and the Units place.
| Position | Hundreds Place | Tens Place | Units Place |
|---|---|---|---|
| Number of Choices | 5 | 5 | 5 |
Step 2: Applying the Fundamental Principle of Counting (Multiplication Principle)
Since the repetition of digits is allowed, the selection of a digit for one position does not restrict the choices for the subsequent positions. According to the Fundamental Principle of Counting, if an operation can be performed in $n$ ways, and a subsequent independent operation can be performed in $m$ ways, then the total number of ways to perform both is $n \times m$.
For a 3-digit number:
- Number of ways to fill the Hundreds place ($n_1$): $5$
- Number of ways to fill the Tens place ($n_2$): $5$
- Number of ways to fill the Units place ($n_3$): $5$
Step 3: Mathematical Calculation
The total number of 3-digit numbers is given by the product of the number of choices for each position:
$Total = n_1 \times n_2 \times n_3$
$Total = 5 \times 5 \times 5$
$Total = 5^3$
$Total = 125$
Final Solution: The total number of 3-digit numbers that can be formed is 125.
More Questions from Class 11 Mathematics Permutations and Combinations EXERCISE 6.1
- 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?
- 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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Other Subjects in CBSE - Class 11
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