WEBVTT

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呢啲睇落可能只係一堆排列整齊既數字

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但事實上佢係數學嘅寶藏

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印度數學家稱為「梅魯火山之梯」

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喺伊朗，佢係「海亞姆三角」

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而喺中國，佢係「楊輝三角」

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喺大部份嘅西方國家

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佢係「帕斯卡三角」

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以法國數學家布萊茲 ‧ 帕斯卡嚟命名

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咁嘅名命睇落有啲唔公平

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因為帕斯卡係後期嘅人
去研究呢款三角形

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但佢嘅貢獻都唔少

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咁到底係咩

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令到世界嘅數學家都咁著迷呢？

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簡單啲嚟講

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係因為佢充滿咗唔同嘅規律同秘密

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首先講下畫呢個三角形嘅方法

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由 1 開始

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想像兩邊各有一個見唔到嘅 0

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將佢哋兩個兩個咁相加

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你就會得到下一行

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重覆咁做

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繼續做，你就會得到呢個三角形

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但其實，帕斯卡三角係無限延伸

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而家，每一行嘅數字

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就係喺二項式 (x+y)^n 展開嘅系數

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而 n 就係行數

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由 0 開始數

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如果 n=2 ，你代入佢

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你會得到 x^2 + 2xy + y^2

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系數，即係變數前嘅數字

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同帕斯卡三角嗰行嘅數字一樣

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當 n=3

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展開之後，你會見到相同嘅情況

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所以呢個三角形係一個

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快捷而且簡單嘅方法去搵呢啲系數

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不過，秘密仲有好多

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例如，將同一行嘅數字加起嚟

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你會得到 2 嘅 n 次方

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或者喺指定嘅一行

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當每個數字都係十進制展開嘅一部份

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即係話

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第三行係 (1x1) + (2x10) + (1x100)

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等於 121，即係 11^2

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睇下如果喺第六行做相同嘅嘢會點？

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一共係 1,771,561，亦即係 11^6

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之後嘅都係咁

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呢三角形仲有唔同嘅幾何應用

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睇下啲對角線

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第一同第二條對角線並唔係好有趣

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全部都係 1 ，同埋正整數

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亦即係自然數

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而喺下一條對角數嘅數字

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我哋稱為三角數

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因為當你將咁多點排列

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你可以排出一個等邊三角形

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喺跟住落嚟嘅對角線上嘅係三角錐體數

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同樣，你可以將呢啲數目砌成三角錐體

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或者咁，遮住所有單數

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當個三角形仲細嘅時候

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你睇唔出係啲咩

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但當你加上成千上萬咁多行之後

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你就會得到一個碎形

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亦即係謝爾賓斯三角形

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呢個三角形唔單只係數學嘅藝術

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佢都幾有用

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特別係計概率同埋組合數學

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例如你想要 5 個小朋友

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而且想知道

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有 3 個女仔同 2 個男仔
呢個理想家庭嘅概率

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喺二項式入面

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呢個即係女仔加男仔嘅 5 次方

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咁我哋睇下第五行

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第一個數字代表 5 個女仔

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而最尾嗰個代表 5 個男仔

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第三個數字就係我哋搵緊嗰個

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呢一行所有可能嘅總和分之 10

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即係 10/32 ，或者 31.25%

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或者，你隨機喺 12 個朋友入面

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揀出一隊 5 人籃球隊

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可以有幾多種組合呢？

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喺組合數學嚟講

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呢個問題可以睇成 12 揀 5

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而且可以用呢條式去計

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或者你可以喺呢個三角形入面

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搵第十二行第六個數字，就會得到答案

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帕斯卡三角形嘅規律

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展現數學優雅交織嘅一面

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我哋至今仍然繼續發現佢新嘅秘密

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例如

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數學家最近發現咗
展開呢種多項式嘅方法

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跟住落嚟我哋會發現啲咩？

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咁就睇你啦