อ้างอิง:
ข้อความเดิมเขียนโดยคุณ -InnoXenT-
56. กำหนดให้ $p,q \in \mathbb{Z}^+$ $\gcd{(p,q)} = 1$ และ
$$\frac{p}{q} = \frac{543}{2549}+\frac{543\times542}{2549\times2548}+...+\frac{543\times542\times ...\times2\times1}{2549\times2548\times ...\times2007}$$
จงหาค่าของ $p+q$
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$$\frac{543}{2549}+\frac{543\times542}{2549\times2548}+\cdots+\frac{543\times542\times \cdots\times2\times1}{2549\times2548\times \cdots\times2007}$$
$$=\frac{543\times542\times \cdots\times2\times1}{2549\times2548\times \cdots\times2007}(\frac{2548\times 2547\times\cdots\times2007}{542\times541\times\cdots\times1}+\frac{2547\times2546\times\cdots\times2007}{541\times540\times \cdots \times1}+\cdots +\frac{2007}{1}+1)$$
$$= \frac{1}{\binom{2549}{543}}(\binom{2548}{542}+\binom{2547}{541}+\cdots+\binom{2006}{0})$$
$$= \frac{1}{\binom{2549}{543}}(\binom{2549}{542})$$
$$=\frac{543}{2007}$$
$$=\frac{181}{669}$$
$$\therefore p+q=850$$
กว่าจะฝันออก ==