To solve the problem of forming a committee of six members that includes one member from the state house of assembly, we can break down the process step-by-step. ### Step 1: Understand the Composition of the Committee We need to form a committee of 6 members from: - 9 state commissioners - 3 members of the state house of assembly Since the committee must include **one member from the house of assembly**, we will first select that member and then choose the remaining members from the state commissioners. ### Step 2: Choose the Member from the House of Assembly We have 3 members in the house of assembly, and we need to choose 1. The number of ways to choose 1 member from 3 is calculated using the combination formula: \[ \text{Number of ways to choose 1 from 3} = \binom{3}{1} = 3 \] ### Step 3: Choose the Remaining Members from State Commissioners After selecting 1 member from the house of assembly, we need to fill the remaining 5 positions of the committee from the 9 state commissioners. The number of ways to choose 5 members from 9 is also calculated using the combination formula: \[ \text{Number of ways to choose 5 from 9} = \binom{9}{5} \] Using the combination formula \(\binom{n}{r} = \frac{n!}{r!(n-r)!}\), we can calculate \(\binom{9}{5}\): \[ \binom{9}{5} = \frac{9!}{5!(9-5)!} = \frac{9!}{5!4!} \] Calculating the factorials: - \(9! = 9 \times 8 \times 7 \times 6 \times 5!\) - \(5! = 120\) - \(4! = 24\) Now substituting back into the formula: \[ \binom{9}{5} = \frac{9 \times 8 \times 7 \times 6}{4 \times 3 \times 2 \times 1} = \frac{3024}{24} = 126 \] ### Step 4: Combine the Choices Now, we combine the number of ways to choose the member from the house of assembly and the number of ways to choose the state commissioners: \[ \text{Total ways} = \text{Ways to choose 1 from house} \times \text{Ways to choose 5 from commissioners} \] \[ \text{Total ways} = 3 \times 126 = 378 \] ### Conclusion Thus, the total number of ways to form the committee, ensuring that it includes one member from the house of assembly, is **378 ways**. ### Explanation of Other Options - **Option B (462 ways)**: This option is incorrect because it likely results from miscalculating the combinations or not properly accounting for the requirement of including one member from the house of assembly. - **Option C (840 ways)**: This option may arise from incorrectly assuming that all members can be chosen without the restriction of including a house member. - **Option D (924 ways)**: This option is also incorrect, possibly due to a misunderstanding of the combination calculations or misapplying the selection criteria. ### Revision Summary - To form a committee of 6 with 1 member from the house of assembly, first choose 1 from 3 house members. - Then choose the remaining 5 from 9 state commissioners using combinations. - The total number of ways to form the committee is 378. - Always ensure to follow the problem's restrictions when calculating combinations.