Question

Ten chairs are numbered as 1 to 10. Three women and two men wish to occupy one chair each. First the women choose the chairs marked 1 to 6, then the men choose the chairs from the remaining. The number of possible ways is

• A. ⁶C₃ × ⁴P₂
• B. ⁶C₃ × ⁴C₂
• C. ⁶P₃ × ⁴C₂
• D. ⁶P₃ × ⁴P₂

Hint:

When solving problems involving selections and arrangements, remember to distinguish between combinations (where order doesn't matter) and permutations (where order does matter). In this case, the women and men are choosing and sitting in specific chairs, so permutations are used.

Answer 1

The Correct Option is D

The correct answer is: (D): \( 6P3 \times 4P2 \)

We are tasked with determining the number of possible ways in which three women and two men can occupy ten numbered chairs under the given conditions:

• The women choose chairs marked 1 to 6.
• The men choose from the remaining chairs.

Step 1: Women choosing their chairs

There are 6 chairs available for the women (chairs marked 1 to 6), and they need to choose 3 of these chairs. The number of ways in which the women can choose 3 chairs from the 6 available chairs is given by a permutation, since the order in which the women sit matters (as each woman will occupy a different chair). The number of ways is:

\[ {}^6P_3 = \frac{6!}{(6 - 3)!} = \frac{6!}{3!} = 6 \times 5 \times 4 = 120 \]

Step 2: Men choosing their chairs

After the women have chosen their chairs, 3 chairs are occupied, leaving 7 chairs (marked 7 to 10) for the men. The men must choose 2 of these remaining chairs. The number of ways in which the men can choose 2 chairs from the remaining 4 chairs is also a permutation, since the order in which the men sit matters. The number of ways is:

\[ {}^4P_2 = \frac{4!}{(4 - 2)!} = \frac{4!}{2!} = 4 \times 3 = 12 \]

Step 3: Total number of possible ways

The total number of possible ways is the product of the two permutations (for the women and the men), so the total number of ways is:

\[ {}^6P_3 \times {}^4P_2 = 120 \times 12 = 1440 \]

Conclusion:
The correct answer is (D): \( 6P3 \times 4P2 \).