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

Question

From a group 7 men and 6 women, five persons are to be selected to form a committee so that at least 3 men are there in the committee. In how many ways can it be done?

Accepted Answer

{"content":[{"content":[{"marks":[{"type":"bold"}],"text":"Step 1 (Identify Cases):","type":"text"},{"text":" We need to choose a 5-person committee from 7 men and 6 women such that there are ","type":"text"},{"marks":[{"type":"bold"}],"text":"at least 3 men","type":"text"},{"type":"text","text":" in the committee. The possible structures for the committee are:"}],"type":"paragraph"},{"content":[{"content":[{"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Case 1: Exactly 3 men and 2 women"}],"type":"paragraph"}],"type":"list_item"},{"content":[{"content":[{"text":"Case 2: Exactly 4 men and 1 woman","type":"text","marks":[{"type":"bold"}]}],"type":"paragraph"}],"type":"list_item"},{"content":[{"type":"paragraph","content":[{"marks":[{"type":"bold"}],"text":"Case 3: Exactly 5 men and 0 women","type":"text"}]}],"type":"list_item"}],"type":"bullet_list"},{"content":[{"marks":[{"type":"bold"}],"text":"Step 2 (Calculate Combinations for each case):","type":"text"}],"type":"paragraph"},{"content":[{"content":[{"content":[{"marks":[{"type":"bold"}],"text":"Case 1 (3 Men, 2 Women):","type":"text"},{"text":" Choose 3 men from 7 and 2 women from 6:","type":"text"}],"type":"paragraph"}],"type":"list_item"}],"type":"bullet_list"},{"attrs":{"latex":"\binom{7}{3} \times \binom{6}{2} = \frac{7 \times 6 \times 5}{6} \times \frac{6 \times 5}{2} = 35 \times 15 = 525"},"type":"math_block"},{"content":[{"content":[{"type":"paragraph","content":[{"marks":[{"type":"bold"}],"text":"Case 2 (4 Men, 1 Woman):","type":"text"},{"text":" Choose 4 men from 7 and 1 woman from 6:","type":"text"}]}],"type":"list_item"}],"type":"bullet_list"},{"attrs":{"latex":"\binom{7}{4} \times \binom{6}{1} = 35 \times 6 = 210"},"type":"math_block"},{"content":[{"content":[{"content":[{"marks":[{"type":"bold"}],"text":"Case 3 (5 Men, 0 Women):","type":"text"},{"type":"text","text":" Choose 5 men from 7:"}],"type":"paragraph"}],"type":"list_item"}],"type":"bullet_list"},{"attrs":{"latex":"\binom{7}{5} = \frac{7 \times 6}{2} = 21"},"type":"math_block"},{"content":[{"marks":[{"type":"bold"}],"text":"Step 3 (Sum the Cases):","type":"text"},{"text":" Total ways = ","type":"text"},{"attrs":{"latex":"525 + 210 + 21 = 756"},"type":"math_inline"},{"text":".","type":"text"}],"type":"paragraph"},{"type":"paragraph","content":[{"text":"Step 4 (Conclusion):","type":"text","marks":[{"type":"bold"}]},{"text":" There are exactly ","type":"text"},{"text":"756","type":"text","marks":[{"type":"code"}]},{"type":"text","text":" distinct ways to form the committee."}]}],"type":"doc"}

Guide / Hint

Solution