Find the number of ways of arranging the letters of the words DANGER, so that no vowel occupies odd

Question

Find the number of ways of arranging the letters of the words DANGER, so that no vowel occupies odd place.

Accepted Answer

{"content":[{"content":[{"text":"Step 1 (Analyze Letters of the Word):","type":"text","marks":[{"type":"bold"}]},{"text":" The word ","type":"text"},{"marks":[{"type":"bold"}],"text":"DANGER","type":"text"},{"text":" consists of 6 distinct letters: D, A, N, G, E, R.","type":"text"}],"type":"paragraph"},{"type":"bullet_list","content":[{"content":[{"content":[{"text":"Vowels: A, E (2 vowels)","type":"text"}],"type":"paragraph"}],"type":"list_item"},{"content":[{"content":[{"text":"Consonants: D, N, G, R (4 consonants)","type":"text"}],"type":"paragraph"}],"type":"list_item"},{"content":[{"content":[{"type":"text","text":"Position indices: 1, 2, 3, 4, 5, 6"}],"type":"paragraph"}],"type":"list_item"},{"type":"list_item","content":[{"content":[{"type":"text","text":"Odd places: 1, 3, 5"}],"type":"paragraph"}]},{"content":[{"type":"paragraph","content":[{"text":"Even places: 2, 4, 6","type":"text"}]}],"type":"list_item"}]},{"type":"paragraph","content":[{"text":"Step 2 (Apply Constraints):","type":"text","marks":[{"type":"bold"}]},{"text":" We are given that ","type":"text"},{"marks":[{"type":"bold"}],"text":"no vowel occupies an odd place","type":"text"},{"text":". This means the 2 vowels (A and E) must occupy the even places (2, 4, or 6).","type":"text"}]},{"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step 3 (Calculate Seating Arrangements):"}],"type":"paragraph"},{"attrs":{"start":1},"content":[{"type":"list_item","content":[{"content":[{"marks":[{"type":"bold"}],"text":"Choose and arrange vowels in even places:","type":"text"},{"text":" We choose 2 even places out of 3 and arrange the 2 vowels:","type":"text"}],"type":"paragraph"}]}],"type":"ordered_list"},{"type":"math_block","attrs":{"latex":"P(3, 2) = 3 \times 2 = 6 \text{ ways}"}},{"attrs":{"start":2},"content":[{"content":[{"content":[{"marks":[{"type":"bold"}],"text":"Arrange consonants:","type":"text"},{"type":"text","text":" The remaining 4 letters (consonants) are arranged in the remaining 4 places:"}],"type":"paragraph"}],"type":"list_item"}],"type":"ordered_list"},{"attrs":{"latex":"4! = 24 \text{ ways}"},"type":"math_block"},{"type":"paragraph","content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step 4 (Calculate Total Product):"},{"text":" Using the product rule:","type":"text"}]},{"attrs":{"latex":"\text{Total Ways} = 6 \times 24 = 144"},"type":"math_block"},{"content":[{"text":"Step 5 (Conclusion):","type":"text","marks":[{"type":"bold"}]},{"text":" The number of ways of arranging the letters is exactly ","type":"text"},{"marks":[{"type":"code"}],"text":"144","type":"text"},{"text":".","type":"text"}],"type":"paragraph"}],"type":"doc"}

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