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Q3:
How many 3-digit even numbers can be made using the digits 1, 2, 3, 4, 6, 7, if no digit is repeated?
Solution :
Step 1: Identifying Constraints and Set Parameters
The objective is to form 3-digit even numbers using the set $S = \{1, 2, 3, 4, 6, 7\}$.
- Total digits available ($n$): 6
- Digits to be selected ($r$): 3
- Condition 1: The number must be even.
- Condition 2: No repetition of digits allowed.
[Per the Fundamental Principle of Counting, if an event can occur in $m$ ways and a second independent event can occur in $n$ ways, the total events occur in $m \times n$ ways.]
Step 2: Analyzing the Units Place (Constraint Application)
For a number to be even, the digit in the units place must be an even number. From the set $S = \{1, 2, 3, 4, 6, 7\}$, the available even digits are $\{2, 4, 6\}$.
Therefore, the units place can be filled in $3$ ways.
Step 3: Filling the Remaining Positions
Having fixed the units place, we select digits for the hundreds and tens places from the remaining digits.
- Total digits originally = 6.
- Digits used in units place = 1.
- Remaining digits = $6 - 1 = 5$.
Hundreds Place: Can be filled from the remaining 5 digits in $5$ ways. [By the Multiplication Principle of Counting].
Tens Place: Can be filled from the remaining 4 digits (as one was used for hundreds and one for units) in $4$ ways.
Step 4: Calculating Total Permutations
The total number of 3-digit even numbers is the product of the possibilities for each position:
$\text{Total} = (\text{Choices for Hundreds}) \times (\text{Choices for Tens}) \times (\text{Choices for Units})$
$\text{Total} = 5 \times 4 \times 3$
$\text{Total} = 60$
Step 5: Tabular Verification of Selection Logic
| Position | Available Options | Count |
|---|---|---|
| Units Place | {2, 4, 6} | 3 |
| Hundreds Place | Total Set - (Units Digit) | 5 |
| Tens Place | Total Set - (Units Digit + Hundreds Digit) | 4 |
Final Solution: The total number of 3-digit even numbers that can be formed is 60.
More Questions from Class 11 Mathematics Permutations and Combinations EXERCISE 6.3
- Q1: How many 3-digit numbers can be formed by using the digits 1 to 9 if no digit is repeated?
- Q2: How many 4-digit numbers are there with no digit repeated?
- Q4: Find the number of 4-digit numbers that can be formed using the digits 1, 2, 3, 4, 5 if no digit is repeated. How many of these will be even?
- Q5: From a committee of 8 persons, in how many ways can we choose a chairman and a vice chairman assuming one person can not hold more than one position?
- Q6: Find $n$ if ${}^{n–1}P_3 : {}^nP_4 = 1 : 9$.
- Q7(i): Find $r$ if (i) ${}^5P_r = 2 \cdot {}^6P_{r-1}$
- Q7(ii): Find $r$ if (ii) ${}^5P_r = {}^6P_{r-1}$.
- Q8: How many words, with or without meaning, can be formed using all the letters of the word EQUATION, using each letter exactly once?
- Q9(i): How many words, with or without meaning can be made from the letters of the word MONDAY, assuming that no letter is repeated, if. (i) 4 letters are used at a time,
- Q9(ii): How many words, with or without meaning can be made from the letters of the word MONDAY, assuming that no letter is repeated, if. (ii) all letters are used at a time,
- Q9(iii): How many words, with or without meaning can be made from the letters of the word MONDAY, assuming that no letter is repeated, if. (iii) all letters are used but first letter is a vowel?
- Q10: In how many of the distinct permutations of the letters in MISSISSIPPI do the four $I$'s not come together?
- Q11(i): In how many ways can the letters of the word PERMUTATIONS be arranged if the (i) words start with P and end with S,
- Q11(ii): In how many ways can the letters of the word PERMUTATIONS be arranged if the (ii) vowels are all together,
- Q11(iii): In how many ways can the letters of the word PERMUTATIONS be arranged if the (iii) there are always 4 letters between P and S?
CBSE Solutions for Class 11 Mathematics Permutations and Combinations
Chapters in CBSE - Class 11 Mathematics
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