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Giải bài 4 trang 80 SGK Toán 10 tập 2 – Chân trời sáng tạo

Xếp 4 viên bi xanh và 5 viên bi trắng có các kích thước khác nhau thành một hàng ngang một cách ngẫu nhiên. Hãy tính số các kết quả thuận lợi cho biến cố:

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Đề bài

Xếp 4 viên bi xanh và 5 viên bi trắng có các kích thước khác nhau thành một hàng ngang một cách ngẫu nhiên. Hãy tính số các kết quả thuận lợi cho biến cố: a) "Không có hai viên bi trắng nào xếp liền nhau". b) "Bốn viên bi xanh được xếp liền nhau".

Lời giải chi tiết

a) Việc xếp 9 viên bi sao cho không có hai viên bi trắng nào xếp liến nhau được thực hiện qua 2 công đoạn: Công đoạn 1: Xếp 4 viên bi xanh trước, vì các viên bi có kích thước khác nhau nên quan tâm đến thứ tự, suy ra công đoạn 1 có 4! = 24 cách. Công đoạn 2: Xếp 5 viên bi trắng vào 5 vị trí xung quanh bi xanh, có quan tâm đến thứ tự nên công đoạn 2 có 5! = 120 cách. Vậy có 120.24 = 2880 kết quả thuận lợi cho biến cố "Không có hai viên bi trắng nào xếp liền nhau". b) Việc xếp 9 viên bi sao cho bốn viên bi xanh được xếp liền nhau được thực hiện qua 2 công đoạn: Công đoạn 1: Xếp 4 viên bi xanh liền nhau, vì các viên bi có kích thước khác nhau nên quan tâm đến thứ tự, suy ra công đoạn 1 có 4! = 24 cách. Công đoạn 2: Coi 4 viên bi xanh liền nhau là 1, cùng với 5 viên bi trắng, tổng là là 6 viên bi hoán đổi vị trí cho nhau nên công đoạn 2 có 6! = 720 cách. Vậy có 24.720 = 17280 kết quả thuận lợi cho biến cố "Bốn viên bi xanh được xếp liền nhau".

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