Arrangement of 12 People in Two Rows with x and y Adjacent

Math Problem Statement

安排包含x和y在內的12個人分坐成兩行,每行有6個人。若x和y須相鄰而坐,求安排坐位的方法數目

這個問題涉及到排列與組合。要求在12個人中，其中兩個特定的人（x和y）必須相鄰而坐，並且要分成兩排，每排6個人。

由於x和y必須相鄰，我們可以將他們視為一個「整體單位」。這樣，我們可以先安排這個「整體單位」以及其餘的10個人（因為兩個人視為一個單位，所以總人數變為11個）。

將這11個單位排列在兩排中，每排6個位置。對於每排的排列，排列的方式為： $\frac{11!}{5!6!} = \binom{11}{6} \times 6! \times 5!$

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Combinatorics
Permutations and Combinations

Number of ways to arrange n items in two groups of size k and (n-k): \binom{n}{k} \times k! \times (n-k)!
Factorial: n! = n × (n-1) × ... × 1

Permutation Theorem
Combination Formula

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