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Q3:
How many 4-letter code can be formed using the first 10 letters of the English alphabet, if no letter can be repeated?
Solution :
Step 1: Identification of Parameters
The problem asks for the number of arrangements (codes) of 4 letters chosen from a set of 10 distinct letters. Since the order of letters in a code matters (e.g., $ABCD \neq DCBA$), this is a permutation problem. [Per the Fundamental Principle of Counting regarding ordered arrangements].
- Total number of available letters ($n$) = $10$
- Number of positions to fill ($r$) = $4$
- Constraint: No letter can be repeated (Sampling without replacement).
Step 2: Mathematical Formulation
The number of permutations of $n$ distinct objects taken $r$ at a time is given by the formula:
$P(n, r) = \frac{n!}{(n - r)!}$
Substituting the given values $n = 10$ and $r = 4$:
$P(10, 4) = \frac{10!}{(10 - 4)!}$
$P(10, 4) = \frac{10!}{6!}$
Step 3: Algebraic Calculation
Expanding the factorial terms to simplify the expression:
$P(10, 4) = \frac{10 \times 9 \times 8 \times 7 \times 6!}{6!}$
Canceling the $6!$ term from the numerator and denominator:
$P(10, 4) = 10 \times 9 \times 8 \times 7$
$10 \times 9 = 90$
$90 \times 8 = 720$
$720 \times 7 = 5040$
Step 4: Logical Verification via Multiplication Principle
Alternatively, by the Multiplication Principle [Per the Fundamental Counting Principle], we have 4 slots to fill:
| Position | Available Options |
|---|---|
| 1st Letter | 10 options |
| 2nd Letter | 9 options (as one is already used) |
| 3rd Letter | 8 options |
| 4th Letter | 7 options |
Total combinations = $10 \times 9 \times 8 \times 7 = 5040$.
Final Solution: The total number of 4-letter codes that can be formed is 5040.
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?
- 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
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