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Giải bài 16 trang 10 sách bài tập toán 10 - Cánh diều

Một tổ có 8 học sinh gồm 4 nữ và 4 nam. Có bao nhiêu cách xếp các học sinh trong tổ:

Tổn✤g hợp đề thi học kì 2 lớp 10 tất cả các môn - Cánh diều

Toán - Văn - Anh - Lí - Hóa - Sinh - Sử - Địa...
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Đề bài

Một tổ có 8 học sinh gồm 4 nữ và 4 nam. Có bao nhiêu cách xếp các học sinh trong tổ: a) Thành một hàng dọc? b) Thành một hàng dọc sao cho nam, nữ đứng xen kẽ nhau?

Phương pháp giải - Xem chi tiết

Bước 1: Tìm số cách xếp n người theo hàng dọc theo hoán vị

Bước 2: Tìm số cách xếp 4 nam (hoặc 4 nữ) trước, khi đó giữa 2 nam (hoặc 2 nữ) tạo thành 1 chỗ trống Bước 3: Sắp xếp 4 nam (hoặc 4 nữ) còn lại vào các chỗ trống được tạo thành rồi sử dụng quy tắc nhân

Lời giải chi tiết

a) Mỗi cách xếp thứ tự vị trí cho 8 học sinh trong tổ là một hoán vị của 8 phần tử.Vậy số cách xếp 8 học sinh trong tổ thành một hàng dọc là: \({P_8} = 8! = 40320\) cách xếpb) Trước tiên ta xếp 4 nữ thành một hàng dọc trước \( \Rightarrow \) Có 4! = 24 cách xếpCứ giữa 2 bạn nữ bất kì tạo thành 1 khoảng trống và có 1 khoảng trống trước bạn nữ đầu tiên hoặc 1 khoảng trống sau bạn nữ cuối cùng \( \Rightarrow \) mỗi trường hợp có tất cả 4 khoảng trốngTH1: Bạn nam đứng đầuTa xếp 4 bạn nam vào 4 khoảng trống (1 khoảng trống trước bạn nữ đầu tiên) thì có 4! cách xếpTH2: Bạn nữ đứng đầuTa xếp 4 bạn nam vào 4 khoảng trống (1 khoảng trống sau bạn nữ cuối cùng) thì có 4! cách xếpVậy có tất cả 4!.4! + 4!.4! = 1 152 cách xếp thỏa mãn

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