From a group of 7 men and 6 women, five persons are to be selected to form a committee so that three

Question

From a group of 7 men and 6 women, five persons are to be selected to form a committee so that three men are there on the committee. In how many ways can it be done?

Accepted Answer

{"content":[{"content":[{"marks":[{"type":"bold"}],"text":"Step 1 (Identify Committee Structure):","type":"text"},{"text":" We want to form a 5-person committee from 7 men and 6 women containing ","type":"text"},{"type":"text","marks":[{"type":"bold"}],"text":"exactly three men"},{"type":"text","text":". This implies:"}],"type":"paragraph"},{"type":"bullet_list","content":[{"content":[{"content":[{"text":"Number of men to choose = 3","type":"text"}],"type":"paragraph"}],"type":"list_item"},{"content":[{"content":[{"text":"Number of women to choose = ","type":"text"},{"attrs":{"latex":"5 - 3 = 2"},"type":"math_inline"}],"type":"paragraph"}],"type":"list_item"}]},{"content":[{"marks":[{"type":"bold"}],"text":"Step 2 (Calculate Combinations):","type":"text"}],"type":"paragraph"},{"content":[{"content":[{"content":[{"type":"text","text":"Ways to choose 3 men from 7:"}],"type":"paragraph"}],"type":"list_item"}],"type":"bullet_list"},{"attrs":{"latex":"\binom{7}{3} = \frac{7 \times 6 \times 5}{3 \times 2 \times 1} = 35"},"type":"math_block"},{"content":[{"content":[{"content":[{"text":"Ways to choose 2 women from 6:","type":"text"}],"type":"paragraph"}],"type":"list_item"}],"type":"bullet_list"},{"attrs":{"latex":"\binom{6}{2} = \frac{6 \times 5}{2 \times 1} = 15"},"type":"math_block"},{"content":[{"marks":[{"type":"bold"}],"text":"Step 3 (Multiply using product rule):","type":"text"},{"text":" The choices are independent, so:","type":"text"}],"type":"paragraph"},{"attrs":{"latex":"\text{Total ways} = \binom{7}{3} \times \binom{6}{2} = 35 \times 15 = 525"},"type":"math_block"},{"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step 4 (Conclusion):"},{"text":" The committee can be formed in exactly ","type":"text"},{"text":"525","type":"text","marks":[{"type":"code"}]},{"type":"text","text":" ways."}],"type":"paragraph"}],"type":"doc"}

Guide / Hint

Solution