This exercise uses the Fundamental Principle of Counting (Multiplication Principle), where the total number of ways to perform a sequence of independent tasks is the product of the number of ways to perform each task.
Class 11 Maths Exercise 6.1 Solutions: Counting Principles, Permutations & Combinations
1. 3-Digit Numbers from Digits 1, 2, 3, 4, 5 (5 Digits Total)
We need to fill three places: $\underline{\text{H}} \quad \underline{\text{T}} \quad \underline{\text{U}}$.
(i) Repetition of the digits is allowed.
Since repetition is allowed, all 5 digits can be used for each place.
- Hundreds place (H): 5 choices (1, 2, 3, 4, 5)
- Tens place (T): 5 choices (1, 2, 3, 4, 5)
- Units place (U): 5 choices (1, 2, 3, 4, 5)$$\text{Total numbers} = 5 \times 5 \times 5 = \mathbf{125}$$
(ii) Repetition of the digits is not allowed.
- Hundreds place (H): 5 choices
- Tens place (T): 4 remaining choices
- Units place (U): 3 remaining choices$$\text{Total numbers} = 5 \times 4 \times 3 = \mathbf{60}$$
2. 3-Digit Even Numbers from Digits 1, 2, 3, 4, 5, 6 (6 Digits Total)
The number must be even, so the Units place (U) is restricted to $\{2, 4, 6\}$ (3 choices). Repetition is allowed.
We fill the places starting with the restriction: $\underline{\text{H}} \quad \underline{\text{T}} \quad \underline{\text{U}}$.
- Units place (U): 3 choices (2, 4, or 6)
- Hundreds place (H): 6 choices (any of the given digits, as repetition is allowed)
- Tens place (T): 6 choices (any of the given digits)$$\text{Total numbers} = 6 \times 6 \times 3 = \mathbf{108}$$
3. 4-Letter Code from First 10 Letters (A-J)
We need to fill four places: $\underline{\text{L1}} \quad \underline{\text{L2}} \quad \underline{\text{L3}} \quad \underline{\text{L4}}$. No repetition is allowed.
- Letter 1 (L1): 10 choices
- Letter 2 (L2): 9 remaining choices
- Letter 3 (L3): 8 remaining choices
- Letter 4 (L4): 7 remaining choices$$\text{Total codes} = 10 \times 9 \times 8 \times 7 = \mathbf{5040}$$
4. 5-Digit Telephone Numbers (Digits 0-9)
The number must start with 67 (two fixed digits) and no digit appears more than once. We have 10 digits available in total.
We need to fill five places: $\underline{D1} \quad \underline{D2} \quad \underline{D3} \quad \underline{D4} \quad \underline{D5}$.
- Digit 1 (D1): 1 choice (must be 6)
- Digit 2 (D2): 1 choice (must be 7)
- Digit 3 (D3): 8 choices (0 to 9, excluding 6 and 7)
- Digit 4 (D4): 7 remaining choices
- Digit 5 (D5): 6 remaining choices$$\text{Total numbers} = 1 \times 1 \times 8 \times 7 \times 6 = \mathbf{336}$$
5. Outcomes of Tossing a Coin 3 Times
The coin has two possible outcomes on each toss: H (Heads) or T (Tails).
- Toss 1: 2 choices (H, T)
- Toss 2: 2 choices (H, T)
- Toss 3: 2 choices (H, T)$$\text{Total outcomes} = 2 \times 2 \times 2 = 2^3 = \mathbf{8}$$(The outcomes are: HHH, HHT, HTH, THH, HTT, THT, TTH, TTT).
6. Signals from 5 Different Coloured Flags
Each signal uses 2 flags, one below the other. The order matters (Flag 1 above Flag 2 is different from Flag 2 above Flag 1). We cannot repeat the flag colour.
We need to fill two places: $\underline{\text{Flag 1}} \quad \underline{\text{Flag 2}}$.
- Flag 1 (Top): 5 choices
- Flag 2 (Bottom): 4 remaining choices$$\text{Total signals} = 5 \times 4 = \mathbf{20}$$