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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?
Solution :
Step 1: Analyzing the Structural Constraints
To form a $3$-digit number, we have three positions to fill: the Hundreds place, the Tens place, and the Units (ones) place. Let the positions be represented by $H$, $T$, and $U$ respectively.
- The available digits are the set $S = \{1, 2, 3, 4, 5, 6\}$.
- Total number of available digits $n = 6$.
- Constraint 1: The number must be even. [By definition, a number is even if its unit digit is divisible by $2$].
- Constraint 2: Repetition of digits is allowed.
Step 2: Determining Permutations for the Units Place ($U$)
For the number to be even, the digit in the units place must be chosen from the set of even digits available in $S$. The even digits in $S$ are $\{2, 4, 6\}$.
Number of choices for the units place ($n_U$) = $3$.
Step 3: Determining Permutations for Remaining Places ($H$ and $T$)
Since repetition is permitted [Per the Fundamental Counting Principle], the choice of one position does not restrict the choices for the others.
- For the Hundreds place ($H$): We can select any of the $6$ digits from $S$. Thus, $n_H = 6$.
- For the Tens place ($T$): We can select any of the $6$ digits from $S$. Thus, $n_T = 6$.
Step 4: Calculation via Fundamental Counting Principle
The total number of ways to form the $3$-digit even number is the product of the number of ways to fill each position [By the Multiplication Principle of Counting]:
$\text{Total Numbers} = n_H \times n_T \times n_U$
$\text{Total Numbers} = 6 \times 6 \times 3$
$\text{Total Numbers} = 36 \times 3 = 108$
Visual Representation of the Choice Tree
Final Solution: 108 even numbers
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?
- 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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